"I don't think he's going to fire Mueller, but I think institutionally it would be nice to have some protections", Graham said Tuesday.
One of the Republican sponsors of the bill, Sen. Harris cited reports that Trump had ordered staff on multiple times to fire Mueller only to be talked out of it by his staff.
The current proposed legislation would give a special counsel recourse through an "expedited judicial review" following a termination.
Senate Democratic Leader Chuck Schumer said it would be "a mistake" not to pass the bill, adding, "We ought to head off a constitutional crisis at the pass, rather than waiting until it's too late".
"It would send a message", said Senator Susan Collins, Republican of ME, who recently endorsed the special counsel proposal written by two Republican senators, Thom Tillis of North Carolina and Lindsey Graham of SC, and two Democratic senators, Chris Coons of DE and Cory Booker of New Jersey.More news: Nishikori Fights Back To Reach Monte Carlo Final
Grassley told the committee that McConnell's views "do not govern what happens here in the Judiciary Committee".
Tillis said that as a former speaker of the North Carolina legislature he would have made the same statement McConnell did if there weren't votes to pass a bill, so it's up to committee members "to get the votes to get it passed".
McConnell told Cavuto that he would be shocked if Trump fired the special counsel, who is investigating contacts between Russian officials and Trump's campaign.
Grassley said that he was unconcerned about McConnell's vow that he wouldn't bring the bill to the floor, saying it was irrelevant to his committee's work.
"I'm the one who decides what we take to the floor, that's my responsibility as the majority leader, we will not be having this on the floor of the Senate", he said.More news: Allison Mack, Smallville actress, charged over Nxivm sex trafficking
Grassley told reporters the bill isn't just about Mueller.
When McConnell refuses to act because he says he doesn't think a threat is real, it means he is happy to let the threat be carried out. "I think most people think we're picking an unnecessary fight with the president".
Connecticut Sen. Richard Blumenthal, a member of the Judiciary Committee, said the argument over constitutionality is a "red herring used by some of my colleagues as a pretext for opposing the bill, when they really have other reasons".
Virginia Sen. Mark Warner, the top Democrat on the Senate intelligence committee, said the special counsel bill is "so much more" than another policy debate. "What I don't trust are future presidents that I don't know yet", he said. He has said in no uncertain terms that he thinks he should be fired.More news: Embiid listed as 'doubtful' for Game 3 vs. Heat