The timeline for next steps on Mr Gaetz’s resolution is variable and it is not yet clear if Democrats would come to Mr McCarthy’s aid.
Mr Gaetz said on Sunday he believed he had enough Republican support to oust Mr McCarthy, 58.
“The only way Kevin McCarthy is speaker of the House at the end of this coming week is if Democrats bail him out,” the Florida representative told CNN.
President Biden has lambasted both Mr McCarthy and the hardliners for failing to live up to an agreement forged around a debt crisis months ago that was meant to avoid a damaging shutdown fight – and for stripping out support for Ukraine.
“Stop playing games, get this done,” Mr Biden said during a press conference on Sunday, adding that he was “sick and tired of the brinksmanship, and so are the American people”.