Republicans are set to kick off their control of the House with a bout of fierce infighting as pro- and anti-McCarthy wings of the GOP conference harden their lines ahead of a key vote to elect the next House speaker on Tuesday.

According to House procedure, if no speaker is elected, then the House will not take up other business until a speaker is chosen, meaning that Republicans could be delayed in starting work on their agenda.

In a letter released January 1, at least nine Republican representatives said they are dissatisfied with Congressman Kevin McCarthy’s leadership and concessions and are withholding their support for him for the moment.

The letter was signed by Representatives Scott Perry, Chip Roy, Paul Gosar, Dan Bishop, Andy Harrus, and Andrew Clyde and Representatives-elect Anna Paulina Luna, Andy Ogles, and Eli Crane.

The representatives write that Mr. McCarthy’s concessions are a “welcome and telling admission of longstanding and deep dysfunction,” but come “almost impossibly late to address continued deficiencies ahead of the opening of the 118th Congress.”

Although the letter did not say that the nine representatives would be guaranteed to oppose Mr. McCarthy on Tuesday, the continued resistance to his leadership does not bode well for his speaker bid.

With at least five other Republican congressmen promising not to support Mr. McCarthy under any circumstances, the Republican leader doesn’t have a clear path to the 218 votes he needs to become speaker.

The resistance to Mr. McCarthy’s speakership comes despite his major concessions, including the restoration of a motion to vacate the chair, which would make Mr. McCarthy easy to oust from the speakership should he ascend to the post.

As conservatives opposing Mr. McCarthy’s leadership dig in their heels, some at the other end of the Republican conference are doubling down on their support for the would-be leader.

A group of 15 members of the 118th Congress, including some from newly flipped districts like Congressman-elect Michael Lawler of New York, have stated that they would not support anyone except Mr. McCarthy.

“Let us be clear: we are not only supporting Kevin McCarthy for Speaker, but are not open to any so-called shadow ‘consensus candidates’ — regardless of how many votes it takes to elect Speaker-designate McCarthy,” a December 29 letter from the group reads.

The Republican conference appears to be at an impasse, a situation that hasn’t arisen since 1923, when it took nine ballots and more than a month of voting to get Speaker Frederick Gillett into the chair.

Such a delay would be an ominous start to the new Republican-controlled House after the party’s disappointing midterm performance. In the letter from those backing Mr. McCarthy, his supporters write that “it is critical that we hit the ground running and stop the dangerous Biden-Pelosi agenda that has made our country less prosperous, less free, and less safe.”

“A protracted floor fight over the Speakership will not only prevent us from doing so, but it will also send the wrong message to the American people at the very moment they entrusted us with governing,” they added.

Congressman Matt Gaetz, one of the leaders of the anti-McCarthy wing, blames the would-be leader for the party’s failure to deliver a larger majority and the likely debacle about to unfold on the floor of the House.

“It is likely that we won’t have a speakership resolved on January 3rd,” Mr. Gaetz told One America News. “That’s driven by Kevin McCarthy’s stubbornness. He doesn’t have the votes — he’s had years to get the votes.”

“For him to persist for a big floor fight that’s messy, well that would be helping the Democrats by his hands not mine,” he added. “I think we need a consensus candidate.”

Representative Bob Good has also redoubled his opposition to Mr. McCarthy’s leadership since the Republican conference meeting Sunday, which was meant to iron out the details of a deal securing his speakership.

“Well what we’ll do is we’ll block Kevin — there’ll be I suspect 10 to 15 members who’ll vote against him on the first ballot tomorrow,” Mr. Good told Fox News. “Then I think you’ll see on the second ballot an increasing number of members vote for a true candidate.”

According to Mr. Good, the coalition opposed to Mr. McCarthy has settled on an alternative to Congressman Andy Biggs, who is his only official opponent, but has refused to name that person. Congressman Fred Upton’s name has been floated as a consensus candidate, as has that of Jim Jordan.

Mr. Biggs reiterated Monday that he is opposed to Mr. McCarthy based on his history as a politician even if Mr. McCarthy were to concede every rules change that Mr. Biggs has requested.