Graham had 39 co-sponsors when he introduced the resolution, but by Friday 50 of the Senate Republican Conference's 53 members had signed on as co-sponsors.

"You know I've been reluctant from the beginning to get involved in the process argument between the White House and the House, but now that the Speaker has scheduled a vote I think that's been overtaken by events," Romney said, asked why he hadn't signed on to Graham's resolution.

Blunt didn't indicate on Monday if he will take it up in committee, instead telling reporters that he is waiting to see the exact language of the House resolution.

"Let's see what she actually proposes," Blunt said. "I read her letter and it could mean not very much, or maybe it will mean more than we're just going to formalize the unfair way we've been doing things."

The resolution cannot currently pass the Senate, where it would need 60 votes. But bringing it to the floor could allow Senate Republicans to formally show their support for Trump.

But the GOP leader, who is co-sponsoring the measure, touted it during his Senate floor speech.