(CNN) — Senate Judiciary Committee Republicans are plowing ahead with their plan to pass a bill out of the committee that would protect special counsel Robert Mueller, despite Majority Leader Mitch McConnell’s vow not to put the measure on the Senate floor.

Senate Judiciary Chairman Chuck Grassley, an Iowa Republican, put the bill on his committee’s agenda for a markup Thursday, which under the panel’s rules means it will it will come before the committee next week.

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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.

“Obviously the views of the majority leader are important to consider, but they do not govern what happens here in the Judiciary Committee,” Grassley said. “If consideration on the floor was the standard for approving a bill in committee or not, we wouldn’t probably be moving any bills out of this committee.”

The bill, which was co-authored by Republican Sens. Thom Tillis of North Carolina and Lindsey Graham of South Carolina and Democratic Sens. Chris Coons of Delaware and Cory Booker of New Jersey, would make it harder for President Donald Trump to fire Mueller by allowing Mueller to seek a 10-day expedited judicial review if he’s dismissed.

The legislation is seen as a potential buffer against concerns that Trump will try to fire Mueller or Deputy Attorney General Rod Rosenstein, who oversees Mueller’s probe.

The measure is on the Judiciary Committee’s agenda for Thursday, but due to the panel’s rules the bill will be marked up the following week.

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Tillis and Graham, the bill’s Republican sponsors, say they think the committee should pass the measure even if McConnell doesn’t plan to put it on the floor.

“The committee is doing what we think needs to be done, and let the leader do what he thinks needs to be done,” Graham said.

Tillis suggested that McConnell could be eventually convinced to change course.

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“It’s on us to get the votes and then talk to the leader,” Tillis said. “If I were the leader I wouldn’t bring a bill to the floor that I didn’t think had a realistic chance of passing.”

McConnell said in a Fox News interview Tuesday that he wouldn’t put the bill on the floor.

“I don’t think he should fire Mueller and I don’t think he’s going to,” the Kentucky Republican said in the interview. “So this is a piece of legislation that isn’t necessary, in my judgment.”

While the bill is bipartisan, a planned amendment from Grassley has Democrats concerned that he’s going to try to undercut the bill, although Republicans say those concerns are unfounded. A Republican aide said the amendment, which hasn’t been released, would add reporting requirements to Congress, including notifications about changes to the probe’s scope and notice of the special counsel’s firing.

Grassley said he showed a copy of the amendment to the bill’s sponsors and California Sen. Dianne Feinstein, the top Democrat on the panel, ahead of Thursday’s meeting, and he planned to provide it to the full committee membership later Thursday.

Feinstein said that some proposed changes appeared to be acceptable, but she still had concerns about requiring law enforcement officials to report prosecutorial decisions during an ongoing investigation.

Leaving the hearing, Feinstein said she was six pages through the 21-page amendment, and wanted to finish reading it before commenting further.

Grassley pushed back on Democratic criticisms that he was trying to undercut the bill or Mueller’s probe. Grassley said some have suggested there’s a “big conspiracy on my part to jump the bill.”

“And I don’t know how to conspire,” he said before gaveling the hearing to a close.

This story has been updated to include additional developments.

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