WASHINGTON — With the federal government one day into a shutdown, the House and Senate reconvened on Saturday for a new round of bitter partisan bickering and public posturing that seemed to cloud the path to a resolution despite initial talk of a compromise.

The shutdown unfolded one year to the day after President Trump’s inauguration, and the political peril it risked for both parties was evident as they traded blame for the crisis.

The Senate met for a rare weekend session at noon — less than 11 hours after it went into recess — and an exasperated-sounding Mitch McConnell, Republican of Kentucky and the majority leader, took the floor. “Well, here we are,” he declared. “Here we are. Day 1 of the Senate Democrats’ government shutdown. We did everything we could to stop them.”

He went on to point a finger at his Democratic counterpart, Senator Chuck Schumer of New York, saying that Mr. Schumer had created an “unfortunate hostage situation and led his party into this untenable position.”

Mr. Schumer, in turn, blamed Mr. McConnell and Mr. Trump.

“He’s turned blowing up bipartisan agreements into an art form,” Mr. Schumer said of the president, adding that “negotiating with President Trump is like negotiating with Jell-O.”

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 brought to the country illegally as children.

The White House is taking a firm stance against entertaining immigration demands while the government is closed.

“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 described Senate Democrats as “obstructionist losers.”

Senator John Kennedy, Republican of Louisiana, during an interview on Sunday in Washington.CreditErin Schaff for The New York Times

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. But Senate Democrats did not immediately get on board with that idea, and it was not clear when a vote on the proposal might be held.

“Something like that is a prescription for trouble,” Senator Ron Wyden, Democrat of Oregon, said Friday night, arguing that a period much shorter than three weeks would encourage decisions on pressing issues.

A bipartisan group of about 18 lawmakers, calling themselves the Common Sense Coalition, met Saturday afternoon in the office of Senator Susan Collins, Republican of Maine, in an effort to find a way forward. Senator Joe Manchin III, Democrat of West Virginia, said the group hoped to present a proposal to Senate leaders either later Saturday or Sunday.

“It’s a travesty if we’re still shut down after tomorrow,” Mr. Manchin said. “It’s a travesty”

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. 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.

Early Saturday morning, 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.”

Shortly after midnight, Senators Mitch McConnell and Chuck Schumer made statements blaming the opposing party for failing to reach an agreement to fund the government.Published OnCreditImage by Erin Schaff for The New York Times

The fate of the young undocumented immigrants known as 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 the Dreamers.

Representative Doug LaMalfa, Republican of California, argued that 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 they’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 closely 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.”