Speaker Kevin McCarthy’s job is on the line as the House returns on Tuesday to confront a funding impasse that could lead to a government shutdown or a challenge to the California Republican’s hold on the top post in the House.

Far-right Republicans are refusing to back a measure to keep the federal government funded past Sept. 30 without substantial spending cuts and stringent new border policies that stand little chance of becoming law. They are also threatening to depose Mr. McCarthy should he turn to Democrats for assistance in scrounging together the votes he needs to avoid a shutdown.

Intensifying the pressure, Representative Matt Gaetz, a Florida Republican and frequent critic of the speaker, was planning to deliver a floor speech on Tuesday outlining the arch-conservative case against Mr. McCarthy, laying the groundwork for a potential move to oust him. The criticism was to cover what Mr. Gaetz and others see as Mr. McCarthy’s failure to live up to promises he made to win the speakership, including his handling of the budget process and ongoing investigations of President Biden and his family.

“Stay tuned,” Mr. Gaetz said on Monday night when reached for comment, declining to elaborate.

Members of the ultraconservative Freedom Caucus were also planning a public event on Tuesday to vent their discontent with the spending landscape.

“The American people expect us to actually fight for them,” Representative Chip Roy, Republican of Texas and a member of the Freedom Caucus, told reporters at the Capitol on Monday. “And speaking as a Texan, I can’t just sit here and rubber-stamp the status quo.”

With Mr. Roy and others railing against plans to temporarily extend funding while the House and the Senate consider yearlong spending bills, Mr. McCarthy faces his most serious leadership test since May, when opposition from the hard right forced him to turn to Democrats to suspend the federal debt ceiling and avoid an economically calamitous default. Right-wing Republicans were irate at the bipartisan compromise, saying the spending levels were far too high.

Mr. McCarthy, who made a series of major concessions to the right wing to win their support to become speaker, faced a brief internal rebellion after the debt deal, but managed to hold on to his post. Now, members of the same faction are warning that he may not survive if he resorts to the same tactics.

“I think there’s a perfect storm brewing in the House in the near future,” Representative Ken Buck, Republican of Colorado, said Sunday on MSNBC.

On the eve of the House’s return, White House officials said on Monday that Mr. Biden would veto a Pentagon spending measure the chamber is to consider this week. The veto threat assailed House Republicans for proposing to cut spending below the levels agreed to in the Fiscal Responsibility Act negotiated this year between the speaker and the president.

“House Republicans had an opportunity to engage in a productive, bipartisan appropriations process,” the White House veto statement said, “but instead, with less than a month before the end of the fiscal year, are wasting time with partisan bills that cut domestic spending to levels well below the F.R.A. agreement and endanger critical services for the American people.”

But the real threat to Mr. McCarthy in navigating the spending fight came not from the White House, but from within his own ranks.

House Republicans are split on the level of spending for both the routine annual spending bills to fund the government through the next year and a temporary measure to avert a shutdown while negotiations over bills continue. In the Senate, Republicans have joined with Democrats in pushing a higher level of spending than House Republicans want, potentially providing the Senate with significant leverage in future negotiations.

Most Senate Republicans are also aligned with Democrats on the need for more financial aid for Ukraine in its war with Russia and want it added to any temporary funding measure. House Republicans have shown growing resistance to such aid.

At the same time, hard-right conservatives are insisting on policy changes such as eliminating funding for the Justice Department’s prosecution of former President Donald J. Trump and new abortion restrictions that have no chance of getting through the Senate or being signed into law by Mr. Biden. Mr. Trump has also urged House Republicans to hold firm against spending without Democratic concessions, a message that resonates with many of them.

“I agree with President Trump that Congress must use the September 30th government funding deadline to cut off funding for Biden’s open border policies and SECURE THE BORDER!” Representative Mary Miller, Republican of Illinois, said on X, formerly Twitter.

And with only four votes to spare on the best of days, Mr. McCarthy may have even less of a margin for error in the coming weeks because of health absences.

While many conservatives dismiss the potential repercussions of a shutdown as minor, Mr. McCarthy has the backing of a significant portion of House Republicans who want to avoid one because of the political consequences as well as the detrimental impact on the public and federal agencies.

Appearing on CNN on Monday, Representative David Joyce, an Ohio Republican who sits on the Appropriations Committee, said Republicans should expect to win only so much in spending talks considering that Democrats control the Senate and the White House. He noted that past shutdowns had backfired on Republicans.

“We have these shutdowns and they haven’t accomplished a hell of a lot,” Mr. Joyce said. “As far as making our government function and making it work for the American people, it doesn’t make a lot of sense.”

But Mr. Roy and like-minded conservatives say the threat of a shutdown is what gives them power at the moment, and he encouraged Mr. McCarthy to work with them to come to an agreement on the spending bills.

“When Kevin works with us to sit down to achieve conservative ends and get 218 Republican votes, we’ve been successful,” Mr. Roy said. “That’s my advice to him in September.”

Efforts to pass individual spending bills stalled in the House in July over internal Republican divisions on spending and policy provisions. Mr. McCarthy hopes to restart the process this week by considering the Pentagon spending bill, which typically draws broad bipartisan support. Democrats are opposing all the House spending measures, however, because they break from the debt limit deal, and it is unclear if the Pentagon legislation has sufficient Republican votes to pass.

In contrast to the House, the Senate on Tuesday was to take up its initial spending bills with debate possibly stretching into next week.

On Monday, Senator Chuck Schumer, Democrat of New York and the majority leader, urged Mr. McCarthy to rebuff the demands of the far right and advance bills on a bipartisan basis as the Senate is doing.

“I implore House Republicans to follow the Senate’s example, reject all-or-nothing tactics, reject unrealistic extreme demands,” Mr. Schumer said. “Don’t let 30 people way out on the extreme dictate what the House does.”

Annie Karni contributed reporting.