Republican Sen. Lindsey Graham of South Carolina urged the White House and Democrats to consider supporting what he called a "breakthrough" compromise on immigration that would allow President Donald Trump to secure his $5 billion demand for border security funds and also codify protections for Deferred Action for Childhood Arrivals (DACA) and Temporary Protected Status (TPS) holders. 

"I don't see Democrats giving us more money, unless they get something," Graham told reporters outside the White House Sunday afternoon after a two-hour lunch with the president.

Graham said he proposed the idea to Trump, who he said called it "interesting." He said the deal would entail the White House supporting the Bridge Act, a bill Graham co-sponsored to grant DACA recipients three-year work permits, as well as legislation to extend the legal status of TPS holders. In exchange, Democrats would need to support a spending measure that includes $5 billion in funding for border security. 

"I think that's in the realm of possibility. The president didn't commit, but I think he's very open minded. I know there's some Democrats out there who would be willing to provide money for a wall, border security if we could deal with the DACA population and TPS people," Graham said. 

A spokesperson for Senate Minority Leader Chuck Schumer, D-N.Y., told CBS News, "At this point, it's clear the White House doesn't know what they want when it comes to border security. While one White House official says they're willing to compromise, another says the president is holding firm at no less than $5 billion for the wall. Meanwhile, the president tweets blaming everyone but himself for a shutdown he called for more than 25 times.

"Senator Schumer has made clear many times publicly and in private to the vice president last week that there are three proposals with funding for smart and effective border security that could pass both chambers – Schumer has repeatedly urged the president to take one of those to end the Trump Shutdown."

Each side is blaming the other for the shutdown. Trump has referred to it on Twitter as the "Schumer Shutdown." The White House has also been targeting presumed incoming House Speaker Nancy Pelosi, D, Calif. The president said in a Dec. 11 Oval Office meeting with Schumer and Pelosi that he'd be "proud to shut down the government for border security," and adding he wouldn't blame Schumer and Pelosi for it.

The Trump administration tried to dismantle DACA, an Obama-era program that shields approximately 800,000 young undocumented immigrants from deportation, in the fall of 2017, but a prolonged court battle has kept the initiative alive. The Department of Homeland Security (DHS) also announced the eventual termination of TPS protections for more than 250,000 immigrants from El Salvador, Honduras, Sudan and Haiti, but a federal judge blocked the decision in early October.

Graham said $5 billion for border security is not enough for either him or the president to support granting DACA recipients a pathway to citizenship.

Trump's rigid stance on border wall funding and Democrats' refusal to budge to his demands have fueled an impasse in budget negotiations on Capitol Hill. The government has been partially closed since December 22 and the shutdown is expected to stretch into the new year, when Democrats take control of the House.

During his remarks, Graham also said the president's plans to begin a full departure of U.S. ground troops from Syria — which the South Carolina senator denounced as "a sign of American weakness" two weeks ago — are "slowing down in a smart way." He said Mr. Trump reassured him that his administration is committed to ensuring that ISIS does not enjoy a resurgence in the Middle Eastern country and that Iran's influence is contained.

"I think we're in a paused situation where we're reevaluating what's the best way to achieve the president's objective of having people pay more and do more," Graham said.