Four Senate Republicans linked arms with Democrats and voted Thursday to advance a bill that would prevent President Trump from being able to fire Special Counsel Robert Mueller, defying the White House and rejecting Mr. Trump’s claims the probe is a “witch hunt.”

The bill cleared the Judiciary Committee on a 14-7 vote over the objections of Republicans who said it was unconstitutional, and Senate Majority Leader Mitch McConnell, who said he won’t bring it to the floor for a vote.

But backers said they hope the strong showing in committee convinces Mr. McConnell to change his mind, and anti-Trump activists began a drumbeat online demanding an immediate vote.

“We have a piece of legislation that I believe will withstand the test of time,” said Sen. Dianne Feinstein, the top Democrat on the committee. “I believe it is constitutional.”

No matter the fate of the bill, the vote stands as a show of confidence for Mr. Mueller, named last year to pursue an investigation into whether the Trump campaign colluded with Russia to sway the election. His probe appears to have expanded well beyond that, and has drawn the repeated ire of Mr. Trump, who has pondered firing him and Deputy Attorney General Rod Rosenstein, who is overseeing the investigation.

The new bill, sponsored by Republican Sens. Lindsey Graham and Thom Tillis and Democratic Sens. Chris Coons and Cory Booker, would give a fired special counsel a chance to challenge the dismissal in court. It would apply to all special counsels going forward, including Mr. Mueller.

Republican Sens. Ben Sasse of Nebraska, Mike Crapo of Idaho, Ted Cruz of Texas, John Cornyn of Texas, John Kennedy of Louisiana, and Mike Lee and Orrin G. Hatch of Utah voted against the bill.

Mr. Lee led the constitutional critique of the bill, saying the president must have the power to fire officials from the Executive Branch, and this bill would trample on that.

“I understand the demand of politics, but that’s no excuse,” he said.

Committee Chairman Chuck Grassley, a Republican, despite voicing his own concerns, backed the bill.

“Congress has an oversight role to play,” he said. “The Supreme Court has recognized as much stating that Congress’s oversight powers are broad.”

Mr. McConnell didn’t comment Thursday on the committee’s bill, but he said last week it would not be given a vote on the Senate floor. House Speaker Paul D. Ryan, Wisconsin Republican, has also said there’s no need for a special counsel bill.

Even if a bill did clear Mr. Trump is certain to veto it, and it’s doubtful Congress could muster the votes to overturn him.

The president has said he has the power to fire Mr. Mueller and Mr. Rosenstein, but has given conflicting answers on whether he’s considering it.

Senate Minority Leader Charles E. Schumer, New York Democrat, said Congress should remove any risk of that happening.

“Rather than waiting for a constitutional crisis, the full Senate should act now,” Mr. Schumer said.

Sen. Thom Tillis, North Carolina Republican, who cosponsored the bill, said his goal wasn’t to target Mr. Trump but to protect all special counsels in the future from presidents.

“I’m trying to create law that has enduring value. It doesn’t reference the special counsel or Special Counsel Mueller,” said Mr. Tillis.

But Mr. Lee shot back and said Mr. Mueller has been discussed throughout the debate over the legislation.

“Let’s not pretend that it’s not,” said Mr. Lee.

The committee did adopt an amendment to the bill, introduced by Mr. Grassley, that requires the attorney general to provide lawmakers with a concluding report, detailing information about decisions to expand or contract the scope of the special counsel’s investigation.

“Transparency and accountability is very much in the public interest,” Mr. Grassley said.

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