Sen. Lindsey Graham, a pivotal member of a bipartisan group of senators trying to overhaul the nation’s immigration system, said on NBC’s Meet the Press Sunday that the key to a bipartisan “grand bargain” on entitlement reform and tax reform “is can we solve immigration?”

If Democrats and Republicans can come up with a bill to overhaul the nation’s immigration system, the South Carolina Republican said, it would open the way to a deal on entitlements and taxes.

Related: Graham warns of North Korean regime overplaying its hand

A member of the Senate Judiciary Committee, Graham has been collaborating with three other Republican senators, Marco Rubio of Florida and Jeff Flake and John McCain of Arizona, along with four Democratic senators, to try to design an immigration bill. This group is known as the Gang of Eight.

“We’re hoping to get this thing done in the next couple of weeks,” the South Carolina Republican told NBC’s David Gregory.

The leading Democratic member of the Gang of Eight, Sen. Charles Schumer of New York said Sunday on CBS's Face the Nation that “So far we're on track. All of us have said there will be no agreement until the eight of us agree to a big, specific bill but hopefully we can get that done by the end of the week.”

The major impediment to reaching a final immigration accord is now the design of a guest worker program, Graham said on Meet the Press. If Republican negotiators are willing to allow a path to citizenship for those foreigners now illegally living in the United States, Graham said, “then the Democratic Party has to give us a guest worker program to help our economy. That’s what we’re arguing over.”

In a message to fellow Republicans, Graham said “the politics of self-deportation are behind us,” – a reference to an idea floated during the 2012 presidential campaign by GOP candidate Mitt Romney. Graham implied that the millions of non-citizens who are illegal living in the United States won’t leave voluntarily and he added that the concept of “self-deportation” was both “impractical” and “offensive.”

Graham said “nuggets” in proposals already leaked from President Barack Obama’s budget plan for the new fiscal year are “somewhat encouraging” and could lead to a deal with Republicans on entitlements and taxes. Obama is “showing some signs of leadership that has been lacking,” he said.

According to Obama administration officials, the president on Wednesday in his Fiscal Year 2014 budget blueprint will propose some changes in entitlement programs – such as a new formula for Social Security, which would effectively reduce retirement benefits, and raising the premiums that upper-income Medicare beneficiaries would need to pay for coverage.

“We’re beginning to set the stage for the grand bargain,” said Graham.

But he mentioned one idea that Obama has not proposed – raising the eligibility age for Medicare benefits from the current age of 65. Graham called for a change to “harmonize the retirement age of Medicare with Social Security.” For middle-aged and younger workers, the eligibility age for full Social Security retirement benefits is 67. For Medicare benefits, the eligibility age is now 65.

On immigration, Graham faces increasingly vocal opposition from some of his fellow Republicans in the Senate.

Related: LGBT activists jump into immigration fray, seeking same-sex partner protections, rights

On Friday, four Republican members of the Judiciary Committee, led by ranking member Sen. Charles Grassley of Iowa, sent a letter to Graham and the other Gang of Eight Republicans saying, “your group has secretly met for months and not consulted with members of the Committee about major changes to our nation’s immigration laws. The time for transparency has come.”

Grassley and his GOP colleagues complained about   the “rushed timetable” which they say Judiciary Committee Chairman Sen. Patrick Leahy, D- Vt., has set for committee approval of the immigration overhaul, moving directly to committee drafting of a bill, with no additional committee hearings.

“We believe it is time for you to discuss the status of your negotiations, disclose what concessions have been made, and provide details to members of the Judiciary Committee as well as the entire Republican Caucus,” the Grassley group said in its letter to Graham and other GOP Gang of Eight members.

Brendan Smialowski / AFP - Getty Images

Senator Lindsey Graham speaks during a press conference on Capitol Hill March 7, 2013.

Sen. Lindsey Graham, a member of the Senate Armed Services Committee, said on NBC’s Meet the Press Sunday that if a military conflict breaks out with North Korea, “the North loses and the South wins with our help.”

Graham said, “I could see a major war happening if the North Koreans overplay their hand this time because the public in South Korea, the United States, and, I think, the whole region is fed up with this guy,” Graham said.

By “this guy,” Graham meant new North Korean leader Kim Jung Un who has been making bellicose threats against the United States in recent days.

Graham said “my biggest fear” is a conflict sparked by the North Korea regime and its misunderstanding of the new thinking in South Korea. The South Koreans, he said, are “not going to put up with this anymore. If there were a South Korean naval vessel sunk this year, any time soon, or shelling of a South Korean island by North Korea, I think the new president of South Korea would be compelled to act. I think the North Koreans are overplaying their hand.”

Graham praised the Obama administration’s handling of the increased tensions with North Korea. “I’m glad we’re not doing the ballistic missile test,” he said, referring to a decision by Secretary of Defense Chuck Hagel to delay an intercontinental ballistic missile test at Vandenberg Air Force Base in California because of the tensions with North Korea. “I’m glad we had the B-2s in the (Korean) theater where they could see them…”

The South Carolina Republican said the Chinese government fears a reunification of North and South Korea. “They don’t want a democratic Korea next to China, so they’re propping up this crazy (North Korean) regime – and they can determine the fate of North Korea better than anybody on the planet,” he said.

Graham also said that the governments of Japan and South Korea haven’t moved to develop their own nuclear weapons capability because “they trust us. As long as South Korea and Japan trust us to be in the fight, they won’t go down the nuclear road. It’s important that they always believe we have their back, it’s important that North Korea knows what happens if they engage anybody in the region associated with us – including with our own troops. They (the North Koreans) lose.” 

Recommended: 'Shared struggle': Gay rights activists jump into immigration fray

Graham, who has just returned from a trip to the Middle East, also commented on the uprising in Syria against the Assad regime.

“The worst is yet to come on Syria if we don’t fix this soon,” he warned. He added that before the United States provides arms to the anti-Assad rebels, it should get a commitment from them to allow an international force to round up and destroy the chemical weapons that Assad’s regime has – “enough weapons to kill millions of people.”

Any new Syrian regime that replaces Bashar al-Assad must, he said, renounce owning chemical weapons.

But the South Carolina Republican warned that “radical elements” in the anti-Assad coalition “are growing by the day.” And it could be “a nightmare in the making” if Assad’s chemical weapons arsenal falls into the hands of radical Islamists in Syria. 

The Senate voted to confirm former Sen. Chuck Hagel as President Barack Obama's next secretary of defense following weeks of dogged opposition by Republican senators to their erstwhile colleague.

The Senate voted 58 to 41 to formally confirm Hagel, on the heels of a procedural vote earlier in the day that cleared the way for Tuesday afternoon's final vote.

That earlier vote dispensed with a filibuster that Senate Republicans had waged for a week and a half against Hagel, whose confirmation was delayed by Republicans past the President's Day recess in order to allow for more time to dig into the former Nebraska senator's background.

A number of Republican detractors — including Sens. John McCain, Ariz., Lindsey Graham, S.C. and Kelly Ayotte, N.H. — reversed their votes on Monday in order to allow the Hagel nomination to move forward.

The Senate voted 71 to 27 to move forward with Hagel's nomination, clearing the 60-vote threshold needed to end the GOP filibuster. A handful of the Republicans who allowed Hagel's nomination to come to a final vote ultimately voted against confirmation.

In the end, Obama was able to win confirmation for Hagel, his choice to succeed outgoing Secretary Leon Panetta at the Pentagon. But not before Republicans were able to drag out the confirmation fight and, in the process, ding Hagel, their onetime GOP Senate colleague from the Cornhusker State.

Republicans had fought strenuously to defeat Hagel, accusing him at points of harboring hostilities toward Israel, and sympathies for the Palestinian militant group Hamas.

Tied into Hagel's nomination as well have been Republicans' long-running effort to ding Obama and his administration over their handling of the Sept. 11, 2012 attacks on a U.S. consulate in Benghazi, Libya. 

Kevin Lamarque / Reuters

Former Senator Chuck Hagel testifies during a Senate Armed Services Committee hearing on his nomination to be Defense Secretary, on Capitol Hill in Washington, in this January 31, 2013, file photo.

"What has their filibuster gained my Republican colleagues?" Senate Majority Leader Harry Reid, D-Nev., asked on the Senate floor. "Twelve days later, Senator Hagel's exemplary record of service to his country remains untarnished."

Reid added: "Senate Republicans have delayed for the better part of two weeks for one reason and one reason only: partisanship."

Hagel didn't necessarily help his cause during a combative confirmation hearing before the Senate Armed Services Committee. Republicans aggressively questioned Hagel on a variety of matters during the Jan. 31 hearing. 

Even still, Democrats held firm in their backing for the former Nebraska senator, helping to move his nomination forward. Republicans, though, managed to buy themselves more time — they said, to more fully investigate Hagel's background — by waging a filibuster against the nomination on Feb. 14. 

Democrats angrily protested the delay, especially as current Defense Secretary Leon Panetta planned to leave the job, as dangerous and unprecedented. Republican opponents of Hagel, though, said at that time that they would drop their objections to holding a confirmation vote after last week's recess.

This story was originally published on Tue Feb 26, 2013 12:37 PM EST

Updated 5:20 p.m. - Senate Republicans managed to achieve their goal of blocking U.N. Ambassador Susan Rice from becoming the next secretary of state after Rice, on Thursday, withdrew her name from consideration by President Barack Obama.

Republicans were more measured in their responses to the withdrawal than they had been in their earlier criticism of Rice, whose prospective nomination had come under fire for her role in publicly explaining the Obama administration’s assessment of the Sept. 11, 2012 attacks on a U.S. consulate in Benghazi, Libya.

"I respect Ambassador Rice’s decision. President Obama has many talented people to choose from to serve as our next Secretary of State," South Carolina Sen. Lindsey Graham said in a statement.

Related: Rice drops out of running for secretary of state

Graham, along with Sens. John McCain, Ariz., and Kelly Ayotte, N.H., had led an effort to pre-empt Obama from naming Rice as the successor to outgoing Secretary of State Hillary Clinton.

"I respect Susan Rice's decision and appreciate her commitment to public service," Ayotte said in a statement. "However, my concerns regarding the terrorist attack in Benghazi go beyond any one individual."

Rice told NBC News in an exclusive interview on Thursday that she no longer wished for Obama to consider her for the position. In a letter to the president, Rice said she feared a confirmation fight in the Senate "would be lengthy, disruptive and costly."  The full interview with Rice will air tonight on Rock Center with Brian Williams at 10 p.m ET.

McCain’s office said: “Senator McCain thanks Ambassador Rice for her service to the country and wishes her well.”

Each of the Republicans, though, expressed continued concern about the Benghazi incident in their statements, and said they would continue their efforts to probe the matter.

Obama said he has accepted Rice's decision, hailing her as an "extraordinarily capable, patriotic, and passionate public servant." He said Rice would continue to serve as U.N. ambassador, and as a member of his national security team.

The trio of Senate Republicans had vowed to work to block Rice's nomination if Obama settled upon the United Nations ambassador as his nominee, stemming from her explanation for the Benghazi attacks. Rice had appeared on public affairs shows the weekend after the attack to assert that the assault -- which left four Americans dead, including Amb. Christopher Stevens – to assert that it was the outgrowth of a spontaneous rally to protest an American video that was offensive to Islam.

An investigation in subsequent weeks revealed that the attack in Benghazi was actually a coordinated terrorist attack, which prompted pointed questions from Republicans about why the administration had first put Rice forth to assert otherwise. Republican presidential nominee Mitt Romney tried for a series of weeks to effectively tarnish Obama politically with the mixed public explanation.

McCain, Ayotte and Graham pressed the matter further after the election, winning a meeting with Rice last month amid speculation that Obama wished to name the trusted adviser to fill the top diplomatic job.

"If Sen. McCain and Sen. Graham and others want to go after somebody, they should go after me," Obama said at a press conference following his re-election to rebuff the Republican troika. "And I'm happy to have that discussion with them. But for them to go after the U.N. ambassador who had nothing to do with Benghazi, and was simply making a presentation based on intelligence that she had received, and besmirch her reputation is outrageous."

But there were indications that the critiques had started to wear on the public perceptions of Rice. In the NBC/WSJ poll released Wednesday, Rice was rated positively by 20 percent of respondents, while 24 percent of said they had a negative perception of her.

TAMPA, Fla. -- South Carolina Sen. Lindsey Graham (R) called on government contractors to put employees on layoff notice before November's election as a way to pressure Congress to address the so-called "fiscal cliff."

Graham, joined by Republican Sens. John McCain (AZ) and Kelly Ayotte (NH), were in Florida for their first stop on a  two-day, four-state tour by these three members of the Senate Armed Services Committee designed to bring attention to the $500 billion in automatic cuts scheduled to begin in January if Congress does not find other ways to cut spending.

“Politicians, you know, quite frankly respond to pressure,” Graham said about the  cuts set to begin in 2013 under the so-called sequestration budget.

“I’m urging every defense industry that could be affected by sequestration to put your employees on notice before November,” he continued. “The more it becomes real to us as to what comes the nation’s way, the more likely we are to solve the problem.”

Graham delivered the remarks inside a University of South Florida auditorium here in Tampa this morning to an audience of military veterans, academics, and defense contractors.

Some in the audience were linked to nearby MacDill Air Force base, a sprawling installation housing the U.S. Central Command, the organization that oversees America’s military activity in the Middle East, including Iraq and Afghanistan.

“There is gridlock in Washington,” McCain said as he warmed the crowd shortly after taking the podium. “I don’t need to tell you that.  It’s hard these days, trying to do the Lord’s work in the city of Satan.”

The line won laughs, but much of the humor today was strictly of the gallows variety.

Before the event began, audience members mingled and expressed satisfaction that South Florida’s defense industry was being recognized.

“I think they’re playing politics with peoples’ lives,” Donna S. Huneycutt, the executive vice president of a small government consulting firm, said of Congress in an interview. 

Huneycutt said she has a staff of 62 people, and nearly had to lay people off last year as a result of earlier budget cuts.

“I’d like to see both sides come to the table and compromise,” she said.

McCain, Graham, and Ayotte called for a bipartisan solution to the crisis.

They signaled they would break with other Republicans and would accept closing loopholes in the tax code in return for concessions from Democrats, including cuts to entitlement programs.

“We shouldn’t put our troops in this position,” Ayotte said. “We shouldn’t put our military feeling like they have the sword of Damocles hanging over their head.”

Ayotte, the wife of a retired Air National Guard pilot who flew combat missions over Iraq, is a buzzed-about prospect for the number-two slot on presumptive Republican presidential nominee Mitt Romney’s ticket and is rumored to be on his short list.

The town hall tour was scheduled to make stops later today in Fayetteville, NC and Norfolk, VA – also home to key military communities.

The tour will wrap Tuesday morning in Merrimack, NH at a facility for the defense contractor BAE Systems.