Originally Published: 04 MAY 21 09:33 ET

Updated: 04 MAY 21 12:50 ET

By Manu Raju and Annie Grayer, CNN

(CNN) -- Rep. Liz Cheney's days as the No. 3 in House GOP leadership appear to be numbered, with speculation growing about her replacement and House Republican Leader Kevin McCarthy contending she has failed to do her job in driving the party's message to take back the majority.

Cheney has grown increasingly isolated within her conference amid her feud with former President Donald Trump, a battle that intensified after she was one of just 10 Republicans who backed his impeachment on a charge of inciting the January 6 insurrection and as she's called out his lie that he actually won the 2020 election.

But on Tuesday, McCarthy claimed her impeachment vote wouldn't cost her the job. Instead, he said that she has not done enough to keep the party unified behind a singular message to win back the majority next year, the clearest sign yet that he might seek to oust her as soon as next week.

"I have heard from members, concerned about her ability to carry out the job as conference chair, to carry out the message," McCarthy told Fox News on Tuesday morning. "We all need to be working as one, if we're able to win the majority. Remember, majorities are not given, they are earned, and that's about the message about going forward."

McCarthy added that the question going forward is "what's our best step forward that we could all work together instead of attacking one another."

But Cheney is showing no signs of backing down amid her growing feud with McCarthy.

"This is about whether the Republican Party is going to perpetuate lies about the 2020 election and attempt to whitewash what happened on Jan 6," Jeremy Adler, Cheney spokesperson, said in a statement. "Liz will not do that. That is the issue."

Cheney has no intention of stepping aside as House Republican conference chair, two people familiar with the matter tell CNN. This means that if she's ousted, it will have to be done through a conference vote.

A vote to oust Cheney from her leadership job seems all but certain and could happen as soon as May 12, according to several senior Republican members and aides. Ultimately, it's McCarthy's call on whether the vote would happen that quickly. But a senior House Republican member predicted to CNN that it's only a matter of time before she's ousted from the leadership post.

"Liz is gone. Just a question of how and when," the GOP member said based on taking the temperature of the conference, but insisted on anonymity to discuss the sensitive issue candidly.

Cheney was not given a heads up before McCarthy spoke on Fox News on Tuesday — and the two haven't spoken since the morning interview, according to a person familiar with the matter.

Cheney has consistently stood by her criticism against Trump even amid the increasing attacks from her own House GOP colleagues. On Monday morning, the Wyoming Republican called out Trump's "BIG LIE" for continuing to make false claims about the 2020 election results. Cheney said later on Monday her party cannot accept the "poison" of the idea that the 2020 election was stolen and should not "whitewash" the January 6 Capitol riot -- and Trump's role in fomenting it, making comments behind closed doors at a conference in Sea Island, Georgia.

Cheney's future as Republican leadership is looking increasingly grim, with chatter now growing in GOP circles about her replacement in leadership as she's become alienated in her conference amid her feud with Trump.

It's still not known which candidates might emerge -- and who would actually win because it'd be a secret ballot election and no candidates have yet announced they would run. Cheney would first have to step aside, or McCarthy could hold a quick vote to force her out, though he hasn't said if he will do that yet.

But with Cheney, the highest-ranking woman in GOP leadership, Republicans are keenly aware how it would appear to replace her with a man. So speculation is growing that a woman might be the best fit for the conference, aides and lawmakers tell CNN.

Some Republican women who might be in the running include Jackie Walorski of Indiana, Elise Stefanik of New York, Maria Elvira Salazar of Florida and Ashley Hinson of Iowa.

Reps. Mike Johnson of Louisiana and Jim Banks of Indiana are also said to be potential candidates.

This story has been updated with additional developments Tuesday.