The House Republican leader, Kevin McCarthy, is fighting for his political life in his quest for the speakership. With the slimmest of majorities and members already coming out in opposition to his candidacy, it may be time to start asking: If not Mr. McCarthy, then who?

In order to become speaker, a House member needs 218 votes, constituting a majority of that body. Given that the GOP now controls 222 seats after this year’s elections, Mr. McCarthy can only afford to lose four of his party members’ votes. With more than a month to go before the official vote, he has already lost three.

Representatives Matt Gaetz, Bob Good, and Ralph Norman have said they will not vote for Mr. McCarthy for speaker. Four others — Representatives Matt Rosendale, Chip Roy, Dan Bishop, and Andy Biggs — have publicly expressed doubts about the GOP leader.

The House vote on the speakership is scheduled for January 3, the first day of the 118th Congress, and no other business will be conducted until that role is filled. If after several rounds of votes Mr. McCarthy cannot get to 218, who else might Republicans turn to?

The most obvious candidate is the current GOP whip, Representative Steve Scalise, though he has not shown any public interest in running for the top job. In a Politico profile, Mr. Scalise said he is happy right where he is, in the no. 2 slot. Mr. McCarthy praised Mr. Scalise for his role in writing the GOP policy plan, known as the “Commitment to America.”

Another candidate could be House Republican Conference leader, Representative Elise Stefanik, who is no. 4 in the GOP House hierarchy. She is close to President Trump and defeated Congresswoman Liz Cheney in claiming the conference leadership. 

Yet Ms. Stefanik would likely face the same resistance from conservative members as Mr. McCarthy. Mr. Roy, who ran against Ms. Stefanik for Republican Conference position after Ms. Cheney was ousted, said at the time that Ms. Stefanik was not conservative enough.

“We should either choose someone who reflects our conservative values, or perhaps leave the position vacant,” Mr. Roy wrote in a letter at the time. He included a list of votes from Ms. Stefanik’s time in Congress that he felt did not reflect a conservative record. 

Should the far-right House Freedom Caucus, which is leading the anti-McCarthy resistance, get its way, moderate Republicans could turn to their Democratic colleagues in an unprecedented power-sharing agreement. One member is already publicly discussing the prospect.

Congressman Don Bacon, a moderate from a district won by President Biden, is a McCarthy supporter who hopes the GOP leader has the requisite votes. “I will support Kevin McCarthy, but if we do get to that point, I do want the country to work and we need to govern,” the Nebraskan said.

Mr. Bacon later hinted at the possibility of working with Democrats to elect a speaker — possibly himself. He told Politico that he has already spoken to “moderate Democrats” about the proposal. 

Considering it takes 218 votes to win the speakership and Democrats will have 212, one could imagine six moderate Republicans banding with the new Democrat leader, Representative Hakeem Jeffries, to make a deal. 

Republican activists are already sounding the alarm about Democrats working to elect a moderate Republican as speaker. One, the founder of a right-wing activist group, Turning Point USA, feels the Freedom Caucus’s tactics are worrisome for Republicans. 

“The Freedom Caucus should be pushing for concessions out of McCarthy, not throwing the speakership to Dems. This is very dangerous and needs to stop,” Charlie Kirk said. 

The problem facing Mr. McCarthy is whether he can satiate his far-right flank while keeping his moderate members in line. If even a small group of far-right members makes demands of Mr. McCarthy that are not palatable to members like Mr. Bacon, he could be in danger of losing the moderate faction. 

In an apparent sign of desperation, Mr. McCarthy’s allies reached out to a moderate Democratic congressman, Henry Cuellar, about switching parties and supporting a McCarthy speakership.

“I told them with all due respect, I will stay as a good conservative Democrat,” Mr. Cuellar said. “That tells me that McCarthy is having a hard time because of the Freedom Caucus.”