Congress careened toward a disruptive government shutdown on Sunday as the Republican-led House groped for a way out of a spending stalemate instigated by the far right just hours before funding for federal agencies would lapse.

Following the resounding defeat of his latest effort to break the impasse on Friday afternoon, Speaker Kevin McCarthy, embattled and grasping to hang on to his job, struggled to find a new way forward that could break through the resistance of a solid bloc of Republicans that has refused to back any stopgap plan that would even temporarily avert the shutdown.

He gathered House Republicans for a private meeting at the Capitol on Saturday to explore a rapidly dwindling set of options.

But the faction showed no sign of relenting, and Mr. McCarthy has so far been unwilling to turn to Democrats, who have repeatedly said they are willing to vote for a bare-bones bill to simply keep the government funded while negotiators can reach a broader agreement on spending for the coming year. His right-wing detractors have said repeatedly they would try to remove the speaker from his post if he did so.

In a reflection of the uncertainty, Republicans who have prided themselves on giving House members 72 hours to read bills before they come to the floor for a vote said in their schedule for Saturday only that the House would consider legislation related to the federal fiscal year that begins on Sunday, with no details on what that legislation might be.

In the Senate, members were set to vote just after midday to advance a bipartisan proposal to fund the government through Nov. 17 while providing $6 billion in assistance to Ukraine and $6 billion for disaster recovery to aid hard-hit Hawaii, Vermont, Florida and other states. It had encountered resistance of its own from Republicans who wanted to add new border security provisions, but that effort had stalled.

With the flow of federal dollars across the government to cease at 12:01 a.m. Sunday with the start of the new fiscal year, it was becoming difficult to see how the two chambers could act and reconcile any differences before then, but Mr. McCarthy was not yet ready to concede that a shutdown was inevitable.

“I don’t think so,” he said Friday evening. “We can still move.”

But he was pressing for the Senate to cut the Ukraine aid, which is opposed by many House Republicans, and said that legislation could not pass the House even though senators of both parties have called the money crucial to demonstrating strong U.S. support for its ally in its fight against Russia. Mr. McCarthy said late Friday on X, formerly known as Twitter, that the Senate measure would be rejected in the House.

“After meeting with House Republicans this evening, it’s clear the misguided Senate bill has no path forward and is dead on arrival,” he wrote.

Still, that legislation could pass with Democratic votes if Mr. McCarthy brought it to the floor and Representative Hakeem Jeffries of New York, the Democratic leader, on Friday urged Mr. McCarthy to move forward with it.

“Everyone in this town knows the bill will pass,” he said.

In an appearance on CNN, Representative Ken Buck, Republican of Colorado, said that Friday’s humiliating defeat of Mr. McCarthy’s proposal represented a vote of “no confidence” in him from the 21 Republicans who joined Democrats in bringing down the bill, a move that severely narrowed the speaker’s options.

“This was a vote where people didn’t have faith that Kevin McCarthy was going to do the right thing,” Mr. Buck said.

Other Republicans said it might be time for their leaders to rethink their strategy of trying to find a way to get the holdouts to vote for what is known as a continuing resolution, or C.R., because they had shown repeatedly that they were not willing to do so.

“I think continuing this path of trying to find a C.R. that they will support is a futile effort, because they will never support a C.R.,” said Representative Mike Garcia of California, a member of the whip team and one of a group of more mainstream Republicans in districts won by President Biden who stand to pay the steepest political price for the shutdown crisis. “Why keep running up a hill that you’re just gonna get shot in the head every time and you’re wasting time and energy? The focus now needs to be getting a C.R. package that can get us to 218. The blend of that voter makeup can change.”

With no resolution in sight, federal agencies were bracing to be shuttered as of Sunday, when programs would cease operation. The armed forces and other so-called essential workers such as air traffic controllers and airport security workers would remain on the job but without pay until the standoff was resolved. Lawmakers see the mid-October deadline for military pay as a consequential deadline were the government to shut down. National parks were to be closed as of Monday, as leaders of both parties said the ramifications would be significant.

“Shutting down the government doesn’t help anybody politically,” Senator Mitch McConnell, Republican of Kentucky and the minority leader, said Friday. “It doesn’t make any meaningful progress on policy. And it heaps unnecessary hardships on the American people as well as the brave men and women who keep us safe.”

Senator Chuck Schumer called on Mr. McCarthy to quit trying to placate the hard-liners among his membership, since any funding deal would ultimately have to be acceptable to Senate Democrats and President Biden.

“At the end of the day, these MAGA extremists, who are the ones responsible for bringing us to the brink, fundamentally do not care about funding the government,” he said. “Some of them are actually gleeful about a shutdown. Coddling the hard right is as futile as trying to nail Jell-O to a wall and the harder the speaker tries, the bigger mess he makes.”