WASHINGTON — Representative Kevin McCarthy of California, the party leader, toiled on Monday — 24 hours before Republicans assume the House majority — to lock down the votes he needs to be elected speaker because he had so far failed to break through entrenched opposition from hard-right lawmakers.

The recalcitrance among ultraconservative lawmakers, even after Mr. McCarthy made a key concession that would weaken his power in the top post, threatened a tumultuous start to the Republican majority in the House. The standoff underscored Mr. McCarthy’s precarious position within his conference and all but guaranteed that even if he eked out a victory he would be a diminished figure beholden to an empowered right flank.

In a vote planned for around midday on Tuesday, when the new Congress convenes, Mr. McCarthy would need to win a majority of those present and voting — 218 if every member of the House were to attend and cast a vote. But despite a grueling weekslong lobbying effort, the California Republican appeared short of the near-unanimity he would need within his ranks to prevail.

A group of five Republicans has publicly vowed to vote against him, and more are quietly opposed or on the fence. Republicans control 222 seats and Democrats are all but certain to oppose him en masse, so Mr. McCarthy can afford to lose only a handful of his own party members.

With little time left ahead of the vote on Tuesday, Mr. McCarthy attempted over the weekend to deliver the hard-liners a major concession by agreeing to a rule that would allow a snap vote at any time to oust the speaker.

Lawmakers opposing him had listed the change as one of their top demands, and Mr. McCarthy had earlier refused to swallow it, regarding it as tantamount to signing the death warrant for his speakership in advance. But in recent days, he signaled he would accept it if the threshold for calling such a vote were five lawmakers rather than a single member.

But that was not enough to sway the five rebels opposing him. and more dissenters emerged on Sunday night, after Mr. McCarthy announced the concession to Republicans in a conference call.

With the holdouts unwilling to bend, Mr. McCarthy could not tell lawmakers and members-elect during the call that he had secured the votes for speaker. Mr. McCarthy could only say that he still had time before the vote on Tuesday, according to two people familiar with the call who insisted on anonymity to describe it.

Roughly two hours later, a separate group of nine conservative lawmakers — most of whom had previously expressed skepticism about Mr. McCarthy’s bid for speaker — derided his efforts to appease their flank of the party as “almost impossibly late to address continued deficiencies.” The group included Representatives Scott Perry of Pennsylvania, the chairman of the Freedom Caucus, and Chip Roy of Texas.

“The times call for radical departure from the status quo — not a continuation of past, and ongoing Republican failures,” the group said in a statement. “For someone with a 14-year presence in senior House Republican leadership, Mr. McCarthy bears squarely the burden to correct the dysfunction he now explicitly admits across that long tenure.”

Mr. McCarthy has pledged to fight for the speakership on the House floor until the very end, even if it requires lawmakers to vote more than once, a prospect that now appears to be a distinct possibility. If he were fail to win a majority on Tuesday, members would take successive votes until someone — Mr. McCarthy or a different nominee — secured enough supporters to prevail.

That could prompt chaos not seen in the House floor in a century. Every speaker since 1923 has been able to clinch the gavel after just one vote.

No viable candidate has yet emerged to challenge Mr. McCarthy, and it was not clear who would be able to unite the fractious Republican Conference if he proved unable to do so. Potential alternatives who could emerge if he fails to secure enough votes include Representative Steve Scalise of Louisiana, his No. 2; Representative Jim Jordan of Ohio, a onetime rival who has strong support among the powerful ultraconservative faction; and Representative Patrick McHenry of North Carolina, one of his close advisers.

Laboring to avoid a scene and cement the speakership, Mr. McCarthy has made a number of concessions over the past few months in attempts to lock up votes of far-right members.

He has called for a “Church-style investigation” into past abuses of power by the Federal Bureau of Investigation and Central Intelligence Agency, a reference to the select committee established in 1975, informally known by the name of the senator who chaired it, Frank Church of Idaho, that looked into abuses by American intelligence agencies.

He toughened his language in response to hard-right demands to oust Alejandro N. Mayorkas, the homeland security secretary, calling on him to resign or face potential impeachment proceedings. He promised Representative Marjorie Taylor Greene of Georgia, who was stripped of her committee assignments for making a series of violent and conspiratorial social media posts before she was elected, a spot on the coveted Oversight Committee.

He threatened to investigate the House select committee investigating the Jan. 6, 2021, attack at the Capitol, promising to hold public hearings scrutinizing the security breakdowns that occurred. Last month he publicly encouraged his members to vote against the lame-duck spending bill to fund the government.

It is unclear whether any single offering from Mr. McCarthy at this point would be enough to win over some lawmakers.

During the call on Sunday, Representative-elect Mike Lawler of New York, who has announced his support for Mr. McCarthy, pointedly asked Representative Matt Gaetz of Florida, a ringleader of the opposition, whether he would vote for Mr. McCarthy if the leader agreed to lower the threshold for a vote to oust the speaker to just one member of Congress. Mr. Gaetz was noncommittal, according to a person on the call who recounted it on the condition of anonymity.

The exchange underscored the challenge Mr. McCarthy faces in attempting to keep control of the House Republican Conference, which includes the task of bargaining with a group of lawmakers who practice a brand of obstructionism that former Representative John A. Boehner, the Ohio Republican who was run out of the speaker post by the far right, famously described as “legislative terrorism.”