"I have never been more depressed about moving forward than right now", said Graham Thursday.
Lawmakers are due back in Washington from their states and congressional districts in the new week.
Mr Trump fired off a series of tweets pushing back against the notion that he does not have a strategy to end what became the longest government shutdown in USA history when it entered its 22nd day on Saturday. But to understand that plan you would have to understand the fact that I won the election, and I promised safety and security for the American people.
With both the Senate and the House adjourned until Monday and a new spending bill nowhere in sight, Trump doubled down on his claim that there's one clear solution to the shutdown: for the new Democratic House of Representatives to give in to his demands to provide funding for his proposed southern border wall.
Graham said he thinks Trump is willing to accept the $5.7 billion he has insisted on for the wall, along with some immigration measures Democrats might find acceptable, such as helping immigrants who were illegally brought to the U.S.as children.
"I would urge him to open up the government for a short period of time, like three weeks, before he pulls the plug".
The shutdown is now the longest in United States history, surpassing the 21-day closure that began during Bill Clinton's presidency in 1995. "They have to get together, forge a compromise, each side has to lose a little bit of face and we've got to get the government back open", he said. "I think we're talking days, not weeks", the chairman and founding member of the conservative House Freedom Caucus said.
Trump has tried on a number of occasions to blame Democrats for the government shutdown, such as when he tweeted in December that "Senator Mitch McConnell should fight for the Wall and Border Security as hard as he fought for anything".
Some 420,000 federal employees whose work is declared essential are working without pay, including the FBI, TSA and other federal law enforcement officers. Graham, who spoke with Trump by telephone on Sunday morning, said the legislative path "is just about shut off" and blamed intransigence by Pelosi. Democrats hold a majority in the house and refuse to include funding for the wall, while Mr Trump refuses to sign off the budget without it.
"If the president was a private citizen, if he was back at the Trump Corporation, he'd be looking at what's going on in Washington and scratching his head".
Last Friday, the partial shutdown was the first time Federal employees did not receive a paycheck, as reported. One notable detail is that, while Democrats are unified in their belief that Trump is to blame for the government shutdown (89 percent feel that way), Republicans are a bit more split (65 percent of Republicans blame congressional Democrats but 23 percent also blame Trump).
California Democrat Adam Schiff, the top Democrat on the House Intelligence Committee, tweeted on Saturday that his panel had tried to get the notes but was blocked by Republicans.
The No. 2 Senate Democrat, Dick Durbin, said yesterday on ABC that "if this president is going to turn to national emergencies every time he disagrees with Congress, I'm against it".
President Donald Trump speaks as he arrives at the White House in Washington, DC, on January 6, 2019, after meetings at Camp David.
The increase in support is sharpest among Republicans, whose backing for Trump's long-standing campaign promise jumped 16 points in the past year, from 71 percent to 87 percent.