“We have yet to reach an agreement on a path forward that would be acceptable for both sides,” the Democratic leader, Senator Chuck Schumer of New York, said after Mr. McConnell’s remarks.
The best hope for a breakthrough appeared to reside with the group of about 20 senators from both parties who met throughout the weekend to try to hammer out a compromise to present to Mr. McConnell and Mr. Schumer.
The group was discussing a plan in which the government would stay open through early February, to be coupled with a promise that the Senate would tackle the issue of immigration in the coming weeks.
“There are, I think, people from both parties of good will who want to have a framework for us to move forward to address all of these issues,” Senator Chris Coons, Democrat of Delaware, said Sunday evening. “But at this point, it is in the hands of leadership, and I really hope they are going to find a way forward.”
A major lingering question was how a compromise might pave the way for passage of legislation to protect the young undocumented immigrants known as Dreamers. Their status is in jeopardy after President Trump moved last year to end an Obama-era program, Deferred Action for Childhood Arrivals, or DACA, that shields them from deportation. Mr. Trump gave Congress until early March to find a resolution to the issue.
For as long as the government is closed, the White House has said it will not entertain demands on immigration. Senators in the bipartisan group proceeded anyway in discussing a compromise in which there would be some kind of promise that the Senate would address the issue in the coming weeks.
“We want to see a commitment to take up immigration,” said Senator Tim Kaine, Democrat of Virginia, “with a belief that we’ll get to a good result for Dreamers.”
Senator Jeff Flake, Republican of Arizona, said it was best for Mr. Trump to let the Senate work out its own solution.
“I just don’t think it helps for him to be involved at all right now,” he said.
Mr. Trump has vacillated on an immigration deal in recent weeks, leaving both sides confused as to what kind of legislation he might accept.
The talk of promised action on immigration also raised other questions, including whether a pledge from Mr. McConnell like the one he offered on Sunday could be a significant enough assurance for Senate Democrats who are worried about the fate of the Dreamers. For instance, a promise of a Senate vote on a stand-alone immigration bill would still leave the possibility that the measure would die in the House, potentially leaving the DACA issue unresolved.
In a reflection of the frustration on both sides, Mr. Trump, who has largely stayed out of negotiations since the government shut down early Saturday, began the day by advising Senate Republicans to use the so-called nuclear option to change Senate rules and cut Democrats out of the process if necessary.
“If stalemate continues, Republicans should go to 51% (Nuclear Option) and vote on real, long term budget, no C.R.’s!” he wrote on Twitter, adding an attack on Democrats.
Leaders from both parties quickly dismissed the idea, but Democrats wasted no time in pointing the finger back at Mr. Trump.
Mr. Schumer’s 11th-hour negotiations with Mr. Trump on Friday have proved to be a focal point for both parties as they have cast blame. On Sunday, Mr. Schumer said that during their meeting at the White House, Mr. Trump had “picked a number” that he wanted to fund a border wall and that Mr. Schumer had accepted in exchange for protections for Dreamers. Hours later, he said, Mr. Trump walked away from a tentative agreement.
“I essentially agreed to give the president something he has said he wants, in exchange for something we both want,” Mr. Schumer said on the Senate floor. “The president must take yes for an answer. Until he does, it’s the Trump shutdown.”
Mr. McConnell has proposed that the temporary spending bill expire on Feb. 8, about a week earlier than the date set in the version of the bill that was passed last week by the House but then failed in the Senate.
The proposed Feb. 8 expiration date was designed to at least partially accommodate a demand by Democrats, who are mindful of preserving what leverage they have, that any temporary funding extension be shorter than the House proposal. But the more significant piece of any possible deal to end the shutdown would be what other strings are attached — particularly when it comes to the politically contentious issue of immigration.
Senators in the bipartisan group had made clear on Sunday that they were eager to bring about an end to the crisis, knowing that they were working against the clock. “Resolution gets more difficult the longer we wait,” said Senator Heidi Heitkamp, Democrat of North Dakota.
Lawmakers in the House, who had been scheduled to recess for the week, were mostly left to watch and wait as their Senate colleagues tried to come together.
Speaker Paul D. Ryan of Wisconsin said on Sunday that House Republicans had agreed to pass Mr. McConnell’s stopgap proposal if it made it out of the Senate. The onus of ending the shutdown, he said, was on Senate Democrats, and he urged them to vote to reopen the government and then restart separate immigration negotiations.
“This is solely done by the Senate Democrats,” Mr. Ryan said on CBS’s “Face the Nation.” “It’s absolutely meaningless. They shut down the government over a completely unrelated issue.”