The hard-right move to oust Speaker Kevin McCarthy from his post has presented House Democrats with a tricky question: Should they help rescue the California Republican who has worked against their agenda and recently opened an impeachment inquiry against President Biden?

Mr. McCarthy’s slender majority and the size of the band of right-wing rebels working to depose him mean that he has little chance of surviving a vote to keep his job — which requires a majority — without at least some support from Democrats.

But it is nearly unheard-of for members of the minority to vote for the opposing party’s candidate for speaker. Top party officials suggest that, given Mr. McCarthy’s multiple partisan transgressions, they would be exceedingly unlikely to do so now without a clear-cut commitment from him to engage Democrats more in governing.

“I am not a cheap date,” said Representative James P. McGovern of Massachusetts, the top Democrat on the Rules Committee, who, like other Democrats, said he had serious trust issues with the speaker.

As they returned to work after a head-spinning Saturday when Congress adopted Mr. McCarthy’s last-minute measure to keep the government open through mid-November, senior Democrats said they had made no firm decision whether they would help him beat back a challenge from Representative Matt Gaetz, Republican of Florida.

Multiple Democrats said that they expected to take a unified position on the fate of the speaker, and that they would meet Tuesday morning to begin the process of determining what they would do.

“Let’s see what happens,” said Representative Steny H. Hoyer, Democrat of Maryland, a former party leader who has had a working relationship with Mr. McCarthy in the past.

Democratic officials said that, absent some sort of tangible and enforceable concession from Mr. McCarthy that benefited them, they could not envision a sufficient number backing him to offset Republican defections.

“I am one Democrat not in much of a mood to help him out,” said Representative Jamie Raskin, Democrat of Maryland. “MAGA Republicans have to solve the problems that MAGA Republicans create.”

Mr. Gaetz made the motion to remove Mr. McCarthy from the speakership on Monday evening, after accusing him of working for Mr. Biden and demanding that he explain any negotiations he had with the White House over the spending bill.

With the critical vote possible as soon as Tuesday, Representative Hakeem Jeffries, Democrat of New York and the minority leader, and his lieutenants have not decided how to proceed.

Their choices include siding with Mr. McCarthy’s right-wing detractors in voting to remove him, joining most Republicans in backing him or helping him in more passive ways: either failing to show up for the vote or voting “present,” both of which would lower the threshold needed for him to win a majority and keep his job.

While the antagonism toward Mr. McCarthy is strong among Democrats, some of them also fear that deposing the speaker in this fashion could both damage the institution and prevent the already nearly paralyzed House from accomplishing much legislatively when lawmakers now face a mid-November deadline to fund the government. They also worry about what, and who, could come next.

One top ally of Mr. McCarthy, Representative Patrick McHenry, Republican of North Carolina, began making the argument Monday that Democrats needed to think long term.

“I think we will have a far more functional House with his speakership going forward than any other person,” Mr. McHenry said.

As of Monday evening, Mr. McCarthy had not approached Democrats with any offer in exchange for their help, and Democrats had not yet reached out to him with an ask, according to people in both camps familiar with the matter and who spoke on the condition of anonymity because they were not authorized to discuss it.

Democrats have been strategizing internally about what concessions they might be able to extract from Mr. McCarthy in exchange for saving him from Mr. Gaetz and his allies. Democrats are eager to see Mr. McCarthy commit to more funding for Ukraine, award more federal projects in Democrats’ districts and honor the deal on spending levels he reached this year with Mr. Biden, according to people familiar with the discussions. All would draw a backlash from Republicans.

Some Democrats have also floated ideas that seem less likely to be acted upon, such as demanding co-chairs for committees, removing right-wing Republicans from the Rules Committee and even obtaining a promise that the G.O.P. campaign committees will not spend money to defeat politically vulnerable Democrats.

Helping Mr. McCarthy is not without political risks for Democrats. Should they go to bat for the speaker, their own voters might not be pleased with an effort to rescue a Republican leader who has allied himself so closely with former President Donald J. Trump and invested so much political capital in tearing down Mr. Biden.

Engaging with Democrats could also backfire on the speaker, giving Republicans already unhappy with him over the spending measure more ammunition to push for his ouster.

Mr. Gaetz practically dared Mr. McCarthy to negotiate across the aisle to save his job. He suggested that Mr. McCarthy had cut a “secret side deal” with Mr. Biden to continue financial support for Ukraine as the country defends itself against a Russian invasion. And he blasted the speaker for partnering with Democrats to keep the government open.

“Working with the Democrats is a yellow brick road that has been paved by Speaker McCarthy,” Mr. Gaetz added.

It is clear that many Democrats are in no rush to save the speaker. Many have grown frustrated with his actions catering to the hard-right members of his party, including his fealty to Mr. Trump and his opening last month of the impeachment inquiry, even in the absence of any evidence of misconduct.

Democrats have a litany of complaints about how they believe they have been wronged by the speaker. Just this past weekend, they accused him of jamming them with a 71-page stopgap bill with no time to review it. Mr. McCarthy then appeared on a Sunday news show portraying Democrats as the party backing a shutdown, when it was mostly Democrats who approved the legislation that averted it.

But their antipathy toward him runs much deeper. While he first blamed Mr. Trump for the events of Jan. 6, 2021, he then flew to Florida to meet with the then-president and to try to mend fences in a public display of allegiance. He also sought to undermine the credibility of the Jan. 6 committee formed when Democrats still had the majority. He negotiated a spending deal with Mr. Biden in the spring and then backed away from it under pressure from his far right. He said he would not begin any impeachment inquiry without a vote on the floor, but then reneged and called for an inquiry when he was facing a rebellion over spending legislation.

“I do not plan to bail out Kevin McCarthy,” Representative Ilhan Omar, the progressive Democrat from Minnesota, said in a statement. “Kevin McCarthy is a speaker who has made it his mission to target women’s ability to make decisions about their bodies, who has made it his mission to cover up a criminal conspiracy from Donald Trump, and is himself a threat to our democracy.”

She said the situation was “a Republican failure,” and it was “on them” if they could not find a speaker they could support.