The likeliest route for lawmakers to reopen the government is to agree on a stopgap spending measure that stretches longer than the few days that Senate Democrats want, but shorter than the four weeks that the House approved on Thursday.
But agreeing on the length of the stopgap bill — essentially, a matter of circling a date on the calendar — is complicated by a number of contentious issues that lawmakers have yet to resolve, particularly the fate of hundreds of thousands of young immigrants, known as Dreamers, brought to the country illegally as children.
In an ominous sign for those who had hoped for a quick resolution, Mr. Trump’s campaign released an ad on Saturday saying that Democrats who stand in the way of cracking down on illegal immigration “will be complicit in every murder committed by illegal immigrants.” And while the government is closed, the White House is taking a firm stance against entertaining immigration demands.
“The president will not negotiate on immigration reform until Democrats stop playing games and reopen the government,” said Sarah Huckabee Sanders, the White House press secretary, who late Friday night described Senate Democrats as “obstructionist losers.”
Mr. McConnell is proposing to shorten the temporary spending bill so that it would expire on Feb. 8 instead of Feb. 16 — an extension of three weeks instead of four. Senate Democrats did not immediately get on board with that idea, but Mr. McConnell said he would move ahead with a procedural vote on the proposal at 1 a.m. Monday, unless Democrats allowed it to be held sooner.
A bipartisan group of roughly 20 senators, who call themselves the Common Sense Coalition, met Saturday afternoon in the office of Senator Susan Collins, Republican of Maine, to forge a compromise. Members said they were working on various proposals to present to Mr. McConnell and Mr. Schumer, with the hope that they would talk to each other.
“We’re going to have to go back and reassess and try to give people a landing spot, a place where everybody feels like they can save a little face and at the same time move forward with getting government running again,” said Senator Michael Rounds, Republican of South Dakota, who participated.
The Senate was scheduled to convene at 1 p.m. Sunday, and the House at 2 p.m. Several senators said they worried that if the shutdown dragged on, people on both sides would dig in, making it harder to come to terms on substantive issues like the fate of the Dreamers.
“There is no defense to what we’re doing,” said Senator Lindsey Graham, Republican of South Carolina, who also participated in the meeting. He added: “I think we look petty. We look that we care more about the party flag than the American flag.”
The shutdown, the first since 2013, took effect after the vast majority of Senate Democrats, as well as a handful of Republicans, voted to block the spending bill that had passed the House. Among that bill’s provisions was a six-year extension of funding for the Children’s Health Insurance Program.
Earlier Friday, Mr. Trump and Mr. Schumer had closed in on an agreement, but those talks eventually fell apart, and Mr. Schumer later blamed the president for backing away from a possible deal.
To reopen the government, at least a dozen or so Senate Democrats will most likely need to agree to any deal, since 60 votes will be required for the measure to clear the Senate. The House would then have to give its approval as well. House members had been scheduled to leave town on Friday for a weeklong recess, but members were advised to remain in Washington, given the possible need to vote on a Senate compromise.
After Friday’s midnight deadline came and went, Mr. Schumer called for the president to sit down with congressional leaders from both parties to work out a deal that would allow the government to open on Monday.
But Republicans, who moved swiftly to brand the crisis as the “Schumer shutdown,” did not seem eager to make concessions, and, in effect, reward Democrats for largely opposing the stopgap bill.
“We do some crazy things in Washington, but this is utter madness,” the House speaker, Paul D. Ryan of Wisconsin, said, adding, “Senate Democrats shut down this government, and now Senate Democrats need to open this government back up.”
Mr. Trump assailed Democrats on Twitter, pointing to the shutdown as a reason that more Republicans need to be elected in the midterm elections this year.
“Democrats are far more concerned with Illegal Immigrants than they are with our great Military or Safety at our dangerous Southern Border,” the president wrote. “They could have easily made a deal but decided to play Shutdown politics instead.”
The fate of the Dreamers is central to the standoff between the two parties. In September, Mr. Trump moved to end an Obama-era program, Deferred Action for Childhood Arrivals, or DACA, that shielded those immigrants from deportation. The president gave Congress until early March to come up with a solution, and Democrats are eager to secure a deal that would protect them.
Representative Doug LaMalfa, Republican of California, argued that Senate Democrats should not “hold 320 million Americans ransom” over helping “a handful of illegal immigrants.” And Senator Thom Tillis, Republican of North Carolina, said his Democratic colleagues had miscalculated their leverage.
“The minority needs to recognize you’re the minority,” Mr. Tillis said. “Any time you come up with a posture that almost suggests that you’re negotiating from the position of the majority of the body, it’s not a very good look.”
But Democrats argued that it was in fact the ineptitude of Republicans — particularly of Mr. Trump, whom they describe as an erratic and unreliable negotiating partner — that had led to the shutdown.
“Despite controlling the House, the Senate and the White House, the Republicans were so incompetent, so negligent, that they couldn’t get it together to keep government open,” Representative Nancy Pelosi of California, the Democratic leader, said on Saturday morning after the House came into session.
As party leaders traded barbs across the Capitol, other lawmakers were clearly growing weary. Senator Lamar Alexander, Republican of Tennessee, noted that he had worked with Mr. Schumer to bring an end to the 2013 shutdown and said no party should ever shut down the government.
“Shutting down the government of the United States of America should never, ever be used as a bargaining chip for any issue, period,” Mr. Alexander said. “It should be to governing as chemical warfare is to real warfare. It should be banned. It should be unthinkable.”