Sen. Lindsey GrahamLindsey GrahamTrump on Ryan snub: 'I don't know what happened' Trump on Jeb: 'I will not say he's low energy' WATCH LIVE: Trump speaks at Nebraska rally MORE (R-S.C.) on Sunday called on his GOP colleagues to take up comprehensive immigration reform.
Graham on CBS’s "Face the Nation" noted his party fared poorly in the election among Hispanic voters.
“I intend to tear this wall down and pass an immigration reform bill that's an American solution to an American problem,” he added.
Graham said it is important to find a way to address the nation's 12 million undocumented immigrants. He said requiring immigrants to have jobs and learn English would be central to approving a path to citizenship in any reform bill.
But any work to tackle immigration reform will have to take a backseat to the "fiscal cliff" as lawmakers focus this lame-duck session on avoiding the combination of looming tax rate rises and automatic spending cuts.
On deficit reduction, Graham urged President Obama to fashion his plan in the style of Simpson-Bowles.
The White House and congressional Republicans are split over the expiring George W. Bush-era tax rates, with Obama calling for them to be extended for families making less than $250,000 per year, while GOP lawmakers want them extended across the board.
Democrats say Obama's reelection shows Americans agree with his plan of calling on the wealthiest earners to pay more in any deficit deal.
Republicans counter that they prefer new revenues from closing tax loopholes and eliminating deductions, along with spending cuts.
“No Republican will vote for higher tax rates,” Graham said.
The president and lawmakers will meet this week to begin hammering out an agreement, and House Speaker John BoehnerJohn BoehnerLobbyists struggle with Trump reality Ryan fans GOP civil war over Donald Trump The Trail 2016: GOP stages of grief MORE (R-Ohio) has said the GOP is open to new revenues in any such deal.