Even though Republicans have yet to officially win a majority in the House of Representatives, the party’s House conference is moving forward with a leadership vote on Tuesday.

The current House minority leader, Representative Kevin McCarthy, is looking like a favorite to cinch the nomination for House speaker if the Republicans succeed in winning the 218 seats needed for a majority.

Ahead of the election it was presumed that Mr. McCarthy, who received President Trump’s endorsement for speaker, would ascend to the position following a “red wave.”

After promising a gain as large as 60 House seats last year and then delivering possibly less than six additional seats, Mr. McCarthy is facing questions about whether he is suitable for the role of speaker.

The most serious challenge so far appears to be coming from the House Freedom Caucus, a group that prevented Mr. McCarthy from winning the speakership in 2015 by refusing to support his candidacy.

The caucus is aiming to secure some rule changes from whoever becomes leader, among them the restoration of the “motion to vacate,” which allows any member to file a privileged motion to force a vote on the speakership.

A former chairman of the House Freedom Caucus, Andrew “Andy” Biggs, is reportedly eyeing the job and considering a challenge to Mr. McCarthy’s leadership.

“I think we need to have a real discussion about whether he should be the speaker or not,” Mr. Biggs told reporters last week. “I think we need to have a very positive statement of what we’re going to accomplish and do, and I haven’t seen that yet.”

The Freedom Caucus, however, appears fractured internally, with one notable member, Representative Marjorie Taylor Greene, saying on Steve Bannon’s War Room podcast that challenges to the current leadership would be a “bad strategy.”

Other House Republicans outside the Freedom Caucus, among them Representative Matthew Gaetz, have also questioned Mr. McCarthy’s ability to lead. 

Mr. Gaetz has said he will oppose Mr. McCarthy’s leadership no matter what concessions are offered, and that he hopes his colleagues will do the same.

Even the current speaker, Representative Nancy Pelosi, took a swing at Mr. McCarthy, telling CNN in reference to his ability to lead: “I don’t think he has it.”

Other members of the press, the public, and elected officials have pointed out that the speaker does not need to be a member of the House. Some Freedom Caucus Republicans have floated the idea of nominating President Trump for the speaker position, according to Representative Jamie Raskin, a Democrat.

Some Democrats in the press — though none in elected office — have suggested that Democrats could cut a deal with Republicans and nominate Representative Liz Cheney as speaker.

This idea immediately drew backlash from both Republicans on the party’s right flank, like Ms. Greene, and Democrats on their party’s left flank, like Representative Nina Turner.

In her interview with Mr. Bannon, Ms. Greene warned supporters about such an eventuality, saying that Democrats were searching for a couple Republican votes to peel off for a speaker vote.

“There are Republicans in our conference right now that would make a deal with the Democrats right now and cross over,” Ms. Greene said. “It’s very very risky right now to produce a leadership challenge.”

This speculation has thrust Republicans with records of compromise — such as Representatives Brian Fitzpatrick, David Valadao, Don Bacon, Mike Garcia, Young Kim, and Beth Van Duyne — into the spotlight. 

The Sun contacted the offices of each of these representatives but received no responses.

Much like in 2015, when opinion writers took to pitching the idea of a bipartisan speakership following Speaker Boehner’s retirement, it is incredibly unlikely to happen.

What is clear is that support for Mr. McCarthy is fractured enough to inspire challenges from the right, and potentially from the center. Mr. Gaetz succinctly summarized the state of play in a tweet ahead of Tuesday’s vote: “Right now if anyone claims to have 218 votes to be Speaker, they are being dishonest.”

Luckily for Mr. McCarthy, he doesn’t need 218 votes just yet. All he needs in Tuesday’s vote is to win a simple majority of his own party’s caucus to win a nomination.