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Washington CNN —  

A bipartisan group of senators introduced legislation Wednesday that would make it harder for special counsel Robert Mueller to be fired for investigating Russian meddling in the 2016 election.

Sens. Lindsey Graham, R-South Carolina, Chris Coons, D-Delaware, Thom Tillis, R-North Carolina, and Cory Booker, D-New Jersey introduced the Special Counsel Independence and Integrity Act, which protect Mueller, including ensure that the special counsel can only be fired for “good cause” by a senior Justice Department official.

The news comes after White House press secretary Sarah Sanders said publicly that President Donald Trump believes he has the power to fire Mueller. She did not suggest the President would be moving to do so.

The legislation also states that if Mueller was fired there would be a 10-day window for Mueller to seek expedited judicial review of his removal to determine whether the firing was for good cause – if it wasn’t, his firing wouldn’t take effect.

The legislation merges previous legislation proposed by the senators last year.

Senate Judiciary Chairman Chuck Grassley has agreed to hold a vote in committee on the new bipartisan bill. Grassley told CNN he wants to hold a committee vote on the bill as soon as Thursday. But any member can hold over the bill and delay a vote for a week, so it’s very possible the vote could get delayed.

But it’s chances of reaching the floor for a vote look slim – GOP leadership is very skeptical about the measure, with Senate Majority Whip John Cornyn renewing his concerns Wednesday morning.

When asked if he would hold a floor vote, Cornyn responded, “I don’t know the answer to that” and said he wanted to read the bill first.

“The biggest question I have is if it did pass, would the President sign it? I think it’s unlikely he would and, as I’ve said, I don’t think it’s necessary,” the Texas Republican said.

Other Republicans, such as Foreign Relations Chairman Bob Corker, said that if such legislation came up, he would support it.

“I have a lot of faith in Mueller and I’ve shared with the President that I think it’d be a tremendous mistake on his part to fire him,” the Tennessee Republican told CNN, adding that firing Mueller “would end his presidency has he knows it.”

Senate Majority Leader Mitch McConnell told reporters on Tuesday he doesn’t believe Mueller would by fired, but he would not say what gives him that confidence when asked repeatedly.

CNN previously reported Trump is considering firing Deputy Attorney General Rod Rosenstein, a move that has gained urgency following the raid of the office of the President’s personal lawyer.

Sen. Dianne Feinstein, the top Democrat on the Senate Judiciary Committee, said she wanted to speak to Grassley and understand exactly what would be considered before agreeing to proceed with the bill this week.

“If there are going to be amendments, we need to see them now and not agree to something we haven’t seen,” Feinstein said.

Republicans are expressing constitutional concerns over the bill, including Sen. John Kennedy of Louisiana, who plans to vote against it in committee.

This story has been updated with additional developments.

CNN’s Manu Raju and Jeremy Herb contributed to this report.