With a potential government shutdown now less than a week away, President Biden and other administration officials this weekend intensified their warnings of the consequences of closing government agencies as they pressed congressional Republicans to find a way out of their spending stalemate.

Both the president and the transportation secretary, Pete Buttigieg, made public calls for Republicans to resolve their differences before next Sunday, when federal funding is set to lapse. They noted that a shutdown would mean that members of the military would go without paychecks, air travelers could experience disruptions and a variety of programs safeguarding the public would be shuttered. Yet even after a weekend of private haggling at the Capitol, there was no sign that the G.O.P. was moving toward a resolution.

“A government shutdown could impact everything from food safety to cancer research to Head Start programs for children,” Mr. Biden said at a Saturday dinner for the Congressional Black Caucus Foundation, blaming the situation “on a small group of extreme Republicans” opposed to a spending deal he cut earlier this year with Speaker Kevin McCarthy. “Now everyone in America could be forced to pay the price.”

“Funding the government is one of the most basic responsibilities of Congress,” he said. “It’s time for Republicans to start doing the job America elected them to do.”

On Sunday, Mr. Buttigieg warned that training for new air traffic controllers would cease during a staffing crunch that has already contributed to travel delays while the working controllers would not be paid.

“They are under enough stress as it is doing that job without having to come into work with the added stress of not receiving a paycheck,” Mr. Buttigieg said on CNN’s “State of the Union” as he made the rounds of Sunday news shows to sound the alarm. “House Republicans need to come to their senses and keep the government running.”

House Republicans gathered on Capitol Hill on Saturday in an effort to chart a path forward this week, but made little progress in coming up with a strategy for overcoming opposition within their own ranks to approving a stopgap spending measure and sending it to the president’s desk in time to keep the government open past next Saturday, the end of the fiscal year.

Instead, after two humiliating procedural defeats on the House floor, Mr. McCarthy relented to demands from the far right to bring to the floor a series of full-year spending bills with steep cuts, though it would be impossible to negotiate final versions with the Senate in the next week. Republicans effectively conceded that the exercise was mostly for show, saying they hoped that advancing the measures would show “good faith” that could ultimately persuade Republican hard-liners to back a measure to keep the government open temporarily.

Mr. McCarthy is now exploring a 45-day extension of federal spending into November, though he is certain to encounter opposition to that timeline even from Republicans who support a stopgap funding measure — and appears to have made little, if any, headway in winning over right-wing lawmakers who have said they have no intention of backing such a bill.

With the House tied in knots, Senator Chuck Schumer, Democrat of New York and the majority leader, has set in motion a procedure for the Senate to pass its own temporary funding measure this week and send it over to the House with both Democratic and Republican votes. A test vote in the Senate is set for Tuesday. A bipartisan group in the House is also exploring procedural options to bring an interim spending plan to the floor.

But if Mr. McCarthy relies on Democrats to pass what is known as a continuing resolution, he is certain to face a challenge to his position from the far right. Appearing on CNN, Representative Tim Burchett, Republican of Tennessee, said that if Mr. McCarthy went in that direction, he would consider voting to oust the speaker.

“That would be something I would look strongly at if we do away with our duty,” said Mr. Burchett, who backs deep spending cuts and has said he would not support stopgap legislation under any circumstances.

At a news conference in the Capitol on Saturday, Representative Garret Graves of Louisiana, a confidant of Mr. McCarthy, argued that Republicans were “ensuring we’re doing everything we can to avoid a government shutdown.”

“We shouldn’t be in a situation where we’re asking our troops to go out there and put their lives on the line and not be paid,” he said. “It would be a failure on our part if we actually reached that point.”

But he acknowledged that passing a stopgap funding measure was not currently the priority of House Republicans, since holdouts have so far made that impossible. Instead, they are pursuing the strategy put forward by Representative Matt Gaetz, Republican of Florida and a McCarthy critic, to first pass appropriations bills on their own.

Still, Mr. Graves insisted that a stopgap funding bill was within the realm of possibility and that members would agree on one in time to stave off a government shutdown.

Even as he told reporters so in the Capitol on Saturday, Representative Matt Rosendale, Republican of Montana and one of the right-wing holdouts, told reporters that he remained a hard “no” on any continuing resolution.

“I won’t support a C.R.,” Mr. Rosendale said as he walked by Mr. Graves issuing his assessment. “I have been consistent on that. I have not changed one bit from that.”

Representative Erin Houchin, Republican of Indiana, responded that “continuing resolution” — a concept that some hard-right lawmakers have made clear they consider unacceptable — was a misnomer for what the party was trying to pass.

“With all due respect to my colleague who said they’re not there yet, what we would be doing is not a continuation. It’s really not a continuing resolution,” Ms. Houchin said. “This is a Republican perspective to stopgap and fund the government while we continue our work.”