But even as the meeting unfolded, the Congressional Budget Office reported that the changes in work requirements for food stamp eligibility — tightening them for some adults, but loosening them for others, including veterans — would actually increase federal spending on the program by $2 billion. Overall, the budget office estimated the deal would make an additional 78,000 people eligible for nutrition assistance.

As Republicans met in the basement of the Capitol, the Rules Committee voted to advance the bill to the House floor on a narrow vote, with two ultraconservative members of the panel bucking their party to oppose allowing the plan to be considered.

With defections from House Republicans stacking up, it remained unclear how many votes Democrats would need to provide to pass the bill and send it to the Senate. Representative Hakeem Jeffries of New York, the Democratic leader, said on Tuesday that Mr. McCarthy had not told him how many Democrats would need to vote for the bill to ensure its passage, but he said Republicans had pledged to produce at least 150 votes for the measure. That would mean several dozen Democrats would have to vote yes to secure passage.

Only one hard-right Republican so far — Mr. Bishop — has publicly said that he considered the debt and spending deal grounds for ousting Mr. McCarthy from his post.

Under the rules House Republicans adopted at the beginning of the year that helped Mr. McCarthy become speaker, any single lawmaker could call for a snap vote to remove him from the speakership, a move that would take a majority of the House. But other hard-right conservatives were holding their fire, saying it was too early to consider the move.

Representative Ken Buck, Republican of Colorado, said on NBC’s “Meet the Press” that he had discussed the issue with the chairman of Freedom Caucus, Representative Scott Perry, Republican of Pennsylvania. “Let’s get through this battle and decide if we want another battle,” Mr. Buck said was the response.

Still, asked if there would be consequences for Mr. McCarthy if the bill passed with more Democratic votes than Republicans, Representative Ralph Norman of South Carolina, replied: “It’s going to be a problem.”