"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 Mr Graham, a South Carolina Republican.
During his presidential campaign, Trump repeatedly pledged that Mexico would pay for the wall, which he says is needed to stem the flow of illegal immigrants and drugs.
Republican senator Lindsey Graham, chairperson of the Senate Judiciary Committee, said he urged the president on Sunday to reopen the government for a limited period to try to get talks going again. Trump said Sunday on Twitter.
Trump fired off a series of tweets pushing back against the notion that he doesn't have a strategy to end what became the longest government shutdown in USA history when it entered its 22nd day Saturday.
As a result, about 800,000 federal employees, including FBI agents and air-traffic controllers, received no pay cheques last Friday. 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).
"How do you negotiate with the speaker of the house when she tells you even if you open up the government, we are not going to give you but $1 for the wall?" "That's why I'm depressed".
Sen. Chris Coons, D-Del., called Graham's idea to reopen the government a "great place to start".
As the shutdown drags on, President Donald Trump and congressional Democrats have not made progress toward any kind of agreement that would put an end to it.
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Most Democrats say they agree on the need for border security, but say there is no national security crisis and believe a wall would be an impractical waste of money. "As a candidate, this is what he talked about, and if there's any mandate he can claim from his election, it was better border security and keeping this nation safe". "I do have a plan on the Shutdown".
Secretary of State Mike Pompeo, travelling Saturday in Abu Dhabi, claimed that morale is good among US diplomats even as many work without pay.
President Trump has visited the US-Mexico border in McAllen, Texas amid a stand off with Congress over getting funding for his border wall. "If we can't at the end of three weeks, all bets are off. See if he can do it by himself through the emergency powers". But the Mexican government has refused and Trump is now demanding that Congress provide funding.
"I think [Schumer] is dealing with the objective reality that his party will not tolerate working with Trump", Gingrich told Catsimatidis.
Opponents say a unilateral presidential move like declaring emergency would be constitutional overreach and set a unsafe precedent in similar controversies.
Both polls found that a majority of Americans oppose Trump's desired wall, as well: 54 percent opposed it in the Post's poll and 56 opposed it in CNN's poll.
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.
Some Republicans on Sunday called for Trump to continue negotiating with Democrats rather than seek to declare a national emergency.