A group of senators failed Wednesday to secure a vote for their bill to protect special counsel Robert Mueller from being fired by President Trump, marking the latest stumble for their efforts to sideline the White House.

Sens. Cory Booker and Chris Coons, both Democrats, and Jeff Flake, a Republican, fear Mr. Trump is angling to fire Mr. Mueller to derail what could be an damaging report on the 2016 elections, Russian meddling and Trump campaign figures’ behavior.

They sought to force to the floor a bill that would have raised the standards for firing Mr. Mueller, making it almost impossible for the president to oust him.

“The bill is becoming more urgent,” Mr. Coons said, pointing to a renewed series of tweets from Mr. Trump this week blasting Mr. Mueller and his team.

But Sen. Mike Lee objected to holding a vote, derailing their effort.

Mr. Lee said setting the special counsel aside in a position that is less answerable to the president would break the separation of powers framework set up by the Constitution, which divides power among the legislature, the judiciary and the executive.

Prosecutors in the federal government are answerable to the president, so carving one out from that scheme would be creating a “de facto fourth branch of government,” Mr. Lee said.

Mr. Mueller is reportedly working to finalize his report, which was launched amid allegations that Trump campaign figures colluded with Russia during the 2016 campaign.

Those accusations have mostly disappeared.

Instead, the president’s opponents now say Mr. Mueller work should continue because of his progress on other matters, including prosecutions of Russian operatives who he charged with trying to meddle in the election, and prosecutions of several Trump campaign figures who’ve been charged with lying to investigators, or with malfeasance from before their time working for the campaign.

While the Mueller protection legislation has some support among Republicans, its chief GOP backer is Mr. Flake, who has said he will vote against all Trump judicial picks until he gets a vote on the Mueller bill.

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