After narrowly avoiding a federal default, the Republican-controlled House and the Democratic-led Senate are now on a collision course over spending that could result in a government shutdown this year and automatic spending cuts in early 2025 with severe consequences for the Pentagon and an array of domestic programs.

Far-right Republicans whose votes will be needed to keep the government funded are demanding cuts that go far deeper than what President Biden and Speaker Kevin McCarthy agreed to in the bipartisan compromise they reached last month to suspend the debt ceiling, but such reductions are all but certain to be nonstarters in the Senate.

The looming stalemate threatens to further complicate a process that was already going to be extraordinarily difficult, as top members of Congress try for the first time in years to pass individual spending bills to fund all parts of the government in an orderly fashion and avoid the usual year-end pileup. If they cannot, under the terms of the debt limit deal, across-the-board spending cuts will kick in in 2025, a worst-case scenario that lawmakers in both parties want to avoid.

The clashes began this week, when House appropriators began considering their spending bills and, working to appease their ultraconservative wing, said they intended to fund federal agencies at below the levels that Mr. Biden and Mr. McCarthy had agreed to.

Democrats balked, saying the move would wreak havoc with the economy and the smooth functioning of government.

“I fully intend to follow the dictates of what we passed in the Senate and the House and what the president signed,” said Senator Patty Murray, Democrat of Washington and the chairwoman of the Appropriations Committee. “I am putting them in their box of chaos,” she said of House Republicans.

The approach was particularly unwise, she added, given that many of the right-wing lawmakers it was aimed at appeasing reflexively vote against government spending bills anyway.

“I don’t believe the country wants us to be there; they don’t want chaos,” Ms. Murray said. “They don’t want a small minority of people to dictate where our economy is going to go.”

Facing a rebellion by hard-right Republicans over the debt limit agreement, Mr. McCarthy and his leadership team blindsided Democrats this week by setting allocations for the 12 annual spending bills at 2022 levels, about $119 billion less than the $1.59 trillion allowed for in the agreement to raise the debt ceiling.

The lower spending levels, demanded by Freedom Caucus members who shut down the House last week to register their ire at the debt limit deal, were pushed through the Appropriations Committee on a party-line vote on Thursday after hours of acrimony during which Democrats accused Republicans of backtracking on the compromise.

“The ink is barely dry on the bipartisan budget agreement, yet we are here to consider the Republican majority’s spending agenda that completely reneges on the compromises struck less than two weeks ago,” said Representative Rosa DeLauro of Connecticut, the top Democrat on the Appropriations Committee.

Representative Kay Granger, Republican of Texas and the committee’s chairwoman, said using the lower number would allow the House to “refocus government spending consistent with Republican priorities.” Mr. McCarthy said that he considered the spending caps established in the agreement simply as a maximum, and that the House wanted to push spending lower.

“There is no limit to how low you could go,” he said, asserting that Republicans wanted to show the public that they could “be more efficient in government, that we can save the hardworking taxpayer more, that we can eliminate more Washington waste.”

But the divergent approaches on either side of the Capitol from the two parties are certain to make passing the spending bills extremely difficult. Failure to pass and reconcile the House and Senate bills by Oct. 1 could lead to a government shutdown. And if the individual bills are not approved by the end of the year, a 1 percent automatic cut would take effect that defense hawks say would be devastating for the Pentagon and U.S. support of Ukraine’s military.

Given the options, those responsible for the spending bills in both chambers say they must move ahead.

“From my perspective, we in the Senate just have to proceed,” said Senator Susan Collins of Maine, the senior Republican on the Appropriations Committee. “I hope that the House will find a way to come to a consensus.”

The four leaders of the two appropriations committees, who for the first time are all women, have said from the start that they wanted to bring the 12 spending bills to the floor under “regular order” and avoid what has become an annual ritual where congressional leaders gather in their suites to cut a last-minute deal lumping hundreds of billions of dollars of spending into one take-it-or-leave-it package.

As part of the debt limit agreement, Senator Chuck Schumer, Democrat of New York and the majority leader, and Senator Mitch McConnell, Republican of Kentucky and the minority leader, issued a statement pledging to “seek and facilitate floor consideration” of the spending bills.

Leaders have avoided floor fights over the spending bills in recent years because they are time-consuming and can force lawmakers to take politically charged votes. But the practice has left many lawmakers complaining that they are being left out of the most basic function of Congress, and the committee heads say they want to bring it to an end.

“What most of us are trying to avoid is a gigantic year-end omnibus that excludes a lot of the rank-and-file members from having input,” Ms. Collins said. “It would be healthy for the dynamic of the Senate, good for our country, and better for federal programs and agencies if we do our work on time.”

At the moment, completing the spending bills on a schedule that has not been met recently looms as a difficult goal to reach with the House and the Senate at odds from the start of the extended review of spending bills. But those in charge say they cannot surrender.

“If we all said, ‘Oh, we can’t do anything, there might be a potential train wreck,’ then why are we here?” Ms. Murray asked. “My job is to get my bills done, to do everything I can to get our bills through the Senate.”

The current turmoil, she said, may dissipate as the deadlines for action approach.

“I wouldn’t take the temperature of where we’re going to be in three months today,” Ms. Murray cautioned. “We’ve got a long ways to go.”