Trump replied: "I think it's the most insulting thing I've ever been asked".
"I would urge him to open up the government for a short period of time, like three weeks, before he pulls the plug", Graham said on "Fox News Sunday". "I'm still hoping we'll have a breakthrough, but right now I don't see one".
The reason why was probably best encapsulated by this, from a GOP aide: "We're used to dealing with rational actors - the idea that once we get into situations like this, one side will see there's no way out and end the stupidity".
On Capitol Hill, there are now no meetings scheduled and no new proposals being traded, aides in both parties say.
More Americans remain opposed to the idea of a border wall than support it, the poll found, although the margin has narrowed over the past year. Such a declaration would enable Trump to bypass congressional approval and build the wall with Pentagon's allocated money.
"The damage done to our Country from a badly broken Border - Drugs, Crime and so much that is bad - is far greater than a Shutdown, which the Dems can easily fix as soon as they come back to Washington!"
President Donald Trump holds a round-table discussion on border security and safe communities with state, local, and community leaders in the Cabinet Room of the White House on January 11.
Trump also suggested that he is open to including protections for "dreamers, " who were brought to the country illegally as children, as part of a deal.
"I think I know this workforce", Johnson said. With nine Cabinet agencies and other offices now shuttered for more than three weeks, Trump on Monday wrote that "Nancy and Cryin' Chuck can end the Shutdown in 15 minutes", referring to House Speaker Nancy Pelosi of California and Senate Minority Leader Charles E. Schumer of NY.More news: Poland arrests Huawei employee on spying charges
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"It's time for those centrists to speak up in their own Republican Senate caucus and tell Mitch McConnell, 'The party's over".
Republican Richard Shelby of Alabama, the chairman of the Appropriations Committee, acknowledged the dilemma, especially as the shutdown continues with no end in sight. Lindsey Graham also says Trump still wants to reach a deal for the wall before agreeing to reopen shuttered government departments.
Trump's comments came in response to reports that an FBI investigation that was opened after Trump fired then-Director James B. Comey in May 2017 included a component to determine whether the president was seeking to help Russian Federation.
About one-quarter of federal government operations have been shut down by a lack of funding since December 22 after Trump demanded $5.7 billion this year from Congress for building a security wall on the southwest U.S. border.
The president also continued to sound off on Democrats who flew to Puerto Rico instead of staying in Washington to find a solution.
Declaring a national emergency over immigration issues is fiercely opposed by Democrats and remains unpopular with some Republicans.
Graham has been among the most outspoken Republicans pushing Trump to use his authority to declare a national emergency to circumvent Congress and build the wall by tapping unspent money sitting in various government accounts, including for military construction and disaster relief. Almost half of all Americans (47 per cent) say there is a serious problem at the border but decline to call it a crisis.
Opponents say a unilateral presidential move like declaring emergency would be constitutional overreach and set a risky precedent in similar controversies.
He now must win concessions from the Democrats, who took over the US House of Representatives this month following November's elections.