Party leaders say they won't agree to fund any kind of wall or barrier between the U.S. and Mexico beyond what's already in place, and Trump insists he won't agree to reopen the government until the wall is funded.
Trump has threatened repeatedly to declare a national emergency to break the stalemate and to order the start of construction of a wall, although on Friday, he retreated from his previously aggressive rhetoric by noting that he is not ready to take such a step now.
Some of Trump's fellow Republicans are warning against a disaster declaration, saying it would undercut Congress's power under the U.S. Constitution to control government spending - and make it easier for a future Democratic president to bypass Capitol Hill. The president initially sounded as though such a move was imminent, but then pulled back. "If we can't at the end of three weeks, all bets are off", Graham said.
A key question is how much more time is Mr Trump willing to give Congress. Mr Graham, who said he and Mr Trump talked by telephone on Sunday morning, said the legislative path "is just about shut off" and blamed Ms Pelosi.
But there was another election, in November, and the outcome of that is that Democrats now control the House and they refuse to give Trump money for a wall. Democrats voted in the past for border security and should again, he said.
Opponents say a unilateral presidential move like declaring emergency would be constitutional overreach and set a risky precedent in similar controversies.
There's good news for Republicans, though, looking to obtain some sort of immigration reform compromise in ongoing budget negotiations: most wall supporters polled by the Washington Post want Republicans - and especially President Trump - to stand firm on making funding for the border wall part of the final budget package. Roughly one-fifth of independents agree with Trump's characterisation, and about half say things are serious but are not a crisis.
"I want to give them the chance to see if they can act responsibly", he told Fox in an interview late last Saturday.
Congressional Democrats are engaged in a more than three-week battle with Senate Republicans and the White House over funding for a wall along the southern border.
The president also said the Democrats don't want to do a deal on the Deferred Action for Childhood Arrivals program that protects young immigrants brought illegally to the United States by their parents.
He tweeted: "I do have a plan on the Shutdown". A person familiar with White House thinking said that in meetings this past week, the message was that the administration is in no rush and wants to consider various options.
Graham and Coons spoke on "Fox News Sunday".
Pelosi argued that Trump is merely trying to steer attention away from special counsel Robert Mueller's investigation and other White House problems.