The vice president faulted the Democrats, declaring it time for them to begin negotiating, but he has essentially blocked potential offramps for the impasse. He has made it clear that Mr. Trump would not drop his insistence on funding for a wall on the southwestern border, which Democrats have branded a nonstarter.
Mr. Pence also indicated that the president was disinclined to accept the idea behind a bipartisan plan that had been under discussion in the Senate — similar to a measure that Republicans and Democrats supported last year, but that the White House rejected — that would trade wall funding for legal status for undocumented immigrants facing the threat of deportation, including the Dreamers and people who previously held Temporary Protected Status.
Privately, he told Mr. Graham’s group that the president would not support the developing proposal, which would have reopened the government for three weeks while Republicans and Democrats worked to hash out a broader legislative deal on the wall and temporary grants of legal status for the two groups.
“We’re kind of stuck,” Mr. Graham conceded.
Speaker Nancy Pelosi, Democrat of California, also showed no signs of budging, urging the Republican-controlled Senate to take up a measure that the House passed on Wednesday to reopen part of the government. The House passed two more measures on Thursday, this time funding the Departments of Housing and Urban Development, Transportation and Agriculture, as well as the Food and Drug Administration.
A dozen Republicans crossed party lines to support one of the measures — slightly more than in previous votes, but no indication that the patience of Mr. Trump’s own party was wearing thin.
“We say to them: ‘Take yes for an answer. This is what you had proposed,’” Ms. Pelosi said at a news conference. “Why are you rejecting it at the expense of the health, safety and well-being of the American people? Do you take an oath to the American people, or to Donald Trump?”
The showdown has forced 800,000 federal workers to go without pay and placed federal benefits for millions more in jeopardy, with the fallout being felt across the United States. Without debate on Thursday, the Senate unanimously passed legislation to ensure that workers who go without salaries receive back pay when the government reopens. Senator Mitch McConnell Republican of Kentucky and the majority leader, said Mr. Trump had assured him he would sign the bill.