Republican leaders in both the House and Senate said Tuesday they will not schedule votes on bipartisan legislation designed to stop President Trump from firing Special Counsel Robert Mueller, saying they don’t think it will come to that.

The comments by House Speaker Paul D. Ryan and Senate Majority Leader Mitch McConnell pour cold water on the effort.

“It’s completely unnecessary,” Mr. Ryan said. “I’ve answered this question pretty much every week. We do not believe that he should be fired, we do not believe that he will be fired. And we therefore don’t think that that is necessary.”

Mr. Ryan said he’s had “conversations” with the White House that have left him confident there won’t be a firing.

Mr. McConnell, speaking with Fox News, sounded similar notes.

“I’m the one who decides what we take to the floor, that’s my responsibility as the majority leader and we will not be having this on the floor of the Senate,” he told Fox host Neil Cavuto.

“That’s not necessary. There’s no indication that Mueller is going to be fired. I don’t think the president’s going to do that. And just a practical matter even if we passed it, why would he sign it?” he said.

A bipartisan group of senators last week introduced a bill that would require a president to show “good cause” to dismiss a special counsel, and that a court review determine whether such good cause existed.

It was prompted by numerous fulminations by Mr. Trump against Mr. Mueller, whom he accuses of running a “witch-hunt,” and Deputy Attorney General Rod Rosenstein, who is charged with overseeing Mr. Mueller.

Democrats questioned Mr. McConnell’s logic, saying if Mr. Trump isn’t going to fire Mr. Mueller, then there’s no harm in passing the bill as a show of support and a guarantee of independence.

Minority Leader Charles E. Schumer said he hoped Mr. McConnell would reconsider his decision.

“While I’m glad the majority leader believes the president would be wrong to fire Special Counsel Mueller, it’s a mistake not to pass legislation to protect the investigation,” the New York Democrat said. “We ought to head off a constitutional crisis at the pass, rather than waiting until it’s too late.”

He said he wanted to see the Judiciary Committee pass the bill even over Mr. McConnell’s objections in order to keep the pressure on.

House Democrats, meanwhile, demanded hearings on Mr. Trump’s decision last week to pardon I. Lewis “Scooter” Libby, the former chief of staff to then-Vice President Dick Cheney, who was convicted of obstruction of justice in a previous special counsel probe.

The Democratic members of the House Judiciary Committee said they believed Mr. Trump was sending a message to anyone thinking of cooperating in Mr. Mueller’s investigation that if they hold out, there could be a pardon in their future as well.

“President Trump’s statements and actions suggest that he has a disturbing view of the pardon power. He seems to believe that he may wield it as a tool to protect those willing to flout the rule of law in the face of a criminal investigation or legal proceeding,” the Democrats said in a letter calling for hearings.

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