The funding impasse has forced the closure of a quarter of government agencies, with some 800,000 federal workers either told to stay home or work without pay until the matter is resolved.
Mr Trump's Republicans had passed a budget bill including $5bn (£4bn) funding for the wall in December, when they still had a majority in the House but they could not get the necessary 60 votes in the 100-seat Senate.
Representatives from the House, including House Speaker-designate Nancy Pelosi, D-Calif.; House Democratic leader-designate Steny Hoyer of Maryland; House Republican leader-designate Kevin McCarthy of California; and House Republican Whip-designate Steve Scalise of Louisiana; along with Senate Democratic leader Charles Schumer of New York; Senate Republican leader Mitch McConnell of Kentucky; Senate Republican Whip-designate John Thune of South Dakota; and Senate Democratic Whip Dick Durbin of IL met with Trump.
Representative Kevin McCarthy, Republican of California and the incoming minority leader, said the group would meet again on Friday, although Democratic officials said they had received no formal invitation to do so. "He could not give a good answer". One would fund the Department of Homeland Security through February 8, providing a month to break the impasse over border security funding, and a second would provide money for the remaining shuttered agencies and departments through September.
From the moment Trump boasted he'd be proud to own the shutdown, Schumer had every political incentive to refuse compromise and spark a shutdown, knowing that Trump would receive the lion's share of the blame.
Hours earlier, Trump charged that Democrats were sacrificing border security for a partisan advantage in the 2020 elections.
Jeffries said when he told his family that, because of his new position, he may be the one to put Pelosi's name forward during the nomination proceeding, his youngest son said to him, "Don't blow the moment, Dad". Representative Jim McGovern of MA, the incoming Democratic Rules chairman, said he was dismayed that any Democrats would oppose the party's own measure for how the House will be run, but he said the internal opposition won't be enough to sink the bill.
There were no raised voices during the meeting, but it was very clear that neither side was moving, the source added.
President Donald Trump speaks during a cabinet meeting at the White House, Wednesday, Jan. 2, 2019, in Washington.
US President Donald Trump made a series of false claims today as he dug in on his demands for border wall funding.
In an unexpectedly long session with the press, he said: "This is national security we're talking about". "The $5 billion approved by the House is such a small amount compared to the level of the problem". That offer was made when Vice President Mike Pence and other top officials met with Schumer at the start of the shutdown.
So far, the administration has rejected a proposal from Democrats to re-open government without money to build a wall along the U.S. -Mexico border.
Democratic and Republican congressional leaders are attending a briefing on border security at the White House Wednesday as the government remains partially shut down. For his part, the President hasn't publicly endorsed that proposal or any other beyond his initial request for $5 billion. Though the House has flipped back to the Democrats, the Republicans increased their majority in the Senate after the midterms.
But the most important finding from the poll is one that points the way forward and out of the government shutdown.
The meeting came as the shutdown dragged through its second week, closing some parks and leaving hundreds of thousands of federal employees without pay. Even after the shutdown ended, it took time for the pay to restart, causing negative ripple effects for months.
None of this bodes well for the task at hand, which is facilitating the expedient conclusion of a shutdown that is having increasingly dire consequences, especially among the most vulnerable Americans.
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