The second-ranking House Republican and the chairman of the Judiciary Committee on Wednesday both announced their campaigns for speaker, setting the stage for a bruising struggle pitting some of the most conservative G.O.P. leaders against one another.

One day after the ouster of Kevin McCarthy as the House’s leader threw the chamber into a state of paralysis, Representative Jim Jordan, Republican of Ohio and a close ally of former President Donald J. Trump, said he would try to become speaker. Mr. Jordan’s bid sets up a challenge with Representative Steve Scalise of Louisiana, currently the No. 2 House Republican, who also announced he would run, and potentially others.

The contenders were all sounding out potential supporters but no real deliberations were to occur before next Tuesday.

The vacancy at the top of the House was creating mounting concerns at the Capitol and the White House about the fate of spending legislation — including hoped-for funding for Ukraine — due in 40 days when it was uncertain when the House would resume functioning and who would be at the helm of the Republican conference.

At the White House, President Biden began his announcement on student loans on Wednesday by addressing chaos in the House and calling on lawmakers to change the “poisonous atmosphere in Washington.”

“We cannot and should not again be faced with an 11th-hour decision or brinkmanship that threatens to shut down the government,” he said. “And we know what we have to do and we gotta we have to get it done in a timely fashion.”

On Capitol Hill, lawmakers also raised alarm about the predicament Congress found itself in

“I want my country to be at work,” said Senator Patty Murray, Democrat of Washington and chair of the Appropriations Committee. “We have a job to do.”

After a historic vote to remove Mr. McCarthy from the speakership on Tuesday, lawmakers quickly departed Washington and scattered to their districts around the country, abandoning the Capitol as Republicans remained deeply divided over who could lead their fractious majority. The sudden departure and the stasis in the House meant that little could be done in Washington even as Congress faced a mid-November deadline to keep the government funded.

“We now find ourselves in a dangerous situation,” said Senator Chuck Schumer, Democrat of New York and the majority leader. “With about 40 days to go before the government shuts down, the House has ground completely to a halt.”

“Until Republicans stop their infighting, the House can vote on no bills,” he said. “No appropriations work can get done. If, God forbid, some national crisis were to occur that demands immediate action, the House would be unable, unable to quickly respond.”

Others noted that even if the House was able to select its new speaker sometime next week, it would take time for that person to ramp up.

“It is going to take a while to get the train back on the tracks,” said Senator Markwayne Mullin, Republican of Oklahoma and a former House member who has close ties to Republicans in that chamber.

The comments underscored the chaos now gripping the chamber, without the ability to conduct legislative business, until a successor to Mr. McCarthy is chosen. The decision by Mr. McCarthy to not seek the post again after being unceremoniously deposed at the hands of the far right touched off a competitive race to succeed him.

Discussions on the future of the conference were being led by Representative Patrick T. McHenry of North Carolina. Mr. McCarthy, a California Republican, had named Mr. McHenry first on a list of potential interim speakers in the event of a calamity or vacancy, but his primary task is to preside over the election of a new speaker. Mr. McHenry has long dismissed the prospect of seeking the top post himself though he could ultimately face a push to draft him if the party is unable to coalesce around another choice.

Mr. Jordan, the politically combative chairman of the House Judiciary Committee, made clear his intentions Wednesday morning as he headed into a meeting of Texas Republicans who were conducting interviews with contenders. Thirty-eight Republicans from the state sit in the House out of 221 total, making their voting bloc crucial in the speaker’s race and giving them leverage.

“I think I can unite the conservative base and the party and the conference, that’s why I’m running,” Mr. Jordan said on his way into lunch with the Texans.

In the past, Mr. Jordan was often at odds with his own party’s leadership but he built an alliance with Mr. McCarthy and when Republicans won the majority in 2022 worked more in concert with the leadership as head of the committee.

Mr. Scalise sent a letter to his colleagues on Wednesday laying out his candidacy and emphasizing his experience as the second in command under Mr. McCarthy.

“I have a proven track record of bringing together the diverse array of viewpoints within our conference to build consensus where others thought it impossible,” said Mr. Scalise. The Louisianian was badly wounded in a 2017 shooting at a baseball practice of congressional Republicans but has steadily recovered. He recently disclosed he was being treated for blood cancer.

Other names that have surfaced as potential contenders include Representative Tom Cole, the Oklahoma Republican and Rules Committee chairman, as well as the No. 3 House Republican, Tom Emmer of Minnesota, who, according to people who have talked with him, is angling for the No. 2 post of majority leader.

Under the current tentative schedule, Republicans intend to hold a party meeting on Tuesday at which the contenders will be able to make their case before their colleagues, with the possibility of picking their choice for speaker for a possible floor vote on Wednesday.

The nominee needs only to secure the majority of the House Republicans to be selected but would have to win a majority of the House, a tall order given the rift among Republicans that made it so difficult for Mr. McCarthy to win the post and do the job for the nine months that he held it. Right-wing Republicans have made clear that they will not support a speaker without assurances that they will see their priorities, including enacting deep spending cuts and severe immigration restrictions, met.

That is nearly impossible to promise given that Democrats control the Senate and the White House. And the situation could be a recipe for further dysfunction on Capitol Hill, most immediately in negotiations on federal spending. The House and Senate must agree by mid-November on the 12 annual appropriations bills to fund the government in the fiscal year that began on Sunday, something that cannot be done without a speaker in place.

Should a new Republican speaker be chosen, the pressure would be immense for that person to push for spending levels far below what Mr. McCarthy had agreed to in a debt deal with President Biden in the spring. Changing the terms of that deal would prompt a clash with the Senate, which is adhering to the agreement.

Catie Edmondson contributed reporting.