Trump walked out of a White House meeting last week with House Speaker Nancy Pelosi and Senate Democratic leader Chuck Schumer, when they refused to approve a border wall, even if he reopened the government and negotiated over border security for the next 30 days.
"We don't want it to come down to a national emergency declaration", said House Republican Whip Steve Scalise of Louisiana.
"Before he pulls the plug on the legislative option, and I think we're nearly there, I would urge him to open up the government for a short period of time, like three weeks, before he pulls the plug, see if we can get a deal", said Graham, a South Carolina Republican.
He has also threatened to invoke a national emergency in order to deploy the military to build the wall, essentially circumventing Congress.
Nonetheless, Trump insisted he had the right to declare an emergency if he chose to, and continued to insist there was a crisis at the border. The president initially sounded as though such a move was imminent, but then pulled back.
Pressed on whether he was confident he could strike a deal with Ms Pelosi, he said he had "no idea".
"Democrats are saying that DACA is not worth it and don't want to include in talks", Trump said Sunday. "See if we can get a deal", Graham, a South Carolina Republican who has called on Trump to invoke emergency action, said on "Fox News Sunday".
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But Trump pointed out that having discussions with world leaders is something that is normal and should not be scrutinized. White House spokeswoman Sarah Sanders said the story was so outrageously inaccurate it didn't even warrant a response.
The speaker's office had no immediate comment. Democrats oppose that step but may be unable to stop it. Many Republicans are wary, too, fearing its use by a future Democratic president. We're all set. It's 100 per cent, and I'll have to do that, ' he said.
The paper reported law enforcement officials became concerned after Mr Trump fired Federal Bureau of Investigation director James Comey in May 2017 and said the agency had to consider whether the president's actions constituted a possible threat to national security.
Meanwhile, Democrats remained resolute that Trump and Republican lawmakers should end the shutdown before rejoining the debate over how to best secure U.S. borders and reform immigration laws.
Many lawmakers spent the weekend in their home states - often hearing constituents' complaints about the shutdown - but will return to Washington on Monday.
Donald Trump went on a morning Twitter rant against media bias and Democrats who refuse to solve the border wall problem as the government shutdown breaks the record as the longest in history. But Trump said he wants to give Congress a chance to negotiate a deal.
Trump added: 'I haven't actually left the White House in months, ' with the exception of a Christmas visit to troops in Iraq and Germany.
'Most conservatives want it to be the last resort he would use, ' said Rep. Mark Meadows, R-N.C., a leader of the conservative House Freedom Caucus who speaks to Trump frequently.
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The White House has been laying the groundwork for an emergency declaration, feared by members of both parties.
Scaramucci said the fact the president made a pledge to build a wall the cornerstone of his campaign explains why he is "obsessed" with the plan now and described the fight as "an ego grudge they have to figure out a way to put aside".
Instead, the president urged lawmakers to provide him the $5.7 billion he is seeking for border security.
"This is where I ask the Democrats to come back to Washington and vote for money for the wall, the barrier", Trump said on Friday.
"I think it's the most insulting article I've ever had written, and if you read the article you'll see that they found absolutely nothing". "Just give in. He's not going to give in", Graham said.
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".
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Senator Ron Johnson, Republican of Wisconsin, said on CNN's "State of the Union" that he'd "hate to see" an emergency declaration by Trump.