WASHINGTON, Nov 13 (Reuters) - U.S. Senate Republicans will gather behind closed doors on Wednesday to pick their next leader in the face of a public pressure campaign by supporters of President-elect Donald Trump who favor the long-shot bid of a hardline Trump loyalist.

Republican senators, who won a majority in last week's election, are due to meet at 9:30 a.m. ET (14:30 GMT) and vote in a secret ballot for a successor to 82-year-old Mitch McConnell, the longest serving party leader in Senate history.

Senators John Thune and John Cornyn, veteran legislators widely seen as institutionalists, have been regarded as frontrunners for much of the behind-the-scenes contest. But a slew of Trump backers ranging from billionaire Elon Musk to conservative commentators Sean Hannity and Tucker Carlson have loudly called for the election of Trump loyalist Rick Scott.

That has made the normally clubby election an early test of Senate independence under Trump, who has not endorsed a candidate but called on the next Republican leader to give him leeway to bypass the normal system of hearings and votes to approve Cabinet nominees. All three contenders quickly signaled their openness to the idea.

Republicans are on track to have a Senate majority of at least 52-48. They also appear poised to retain a majority in the House of Representatives, which would give Trump a governing trifecta in national politics.

But some of Trump's loudest supporters have expressed concern that Thune and Cornyn, who have both worked closely with McConnell, might lack the willingness to deliver on some of Trump's campaign promises.

"Without Rick Scott, the entire Trump reform agenda wobbly," Trump adviser Robert F. Kennedy Jr said in a post on X.

Multiple Senate Republicans have pushed back against the idea of simply voting to back Scott because of public pressure. The job of majority leader, they said, requires someone who has spent time in building trust and support within the caucus.

Some of Scott's supporters say he is unlikely to win without a Trump endorsement, setting off a contest between public calls for the president-elect to back Scott and private urgings from Senate Republicans for him to stay out of the race.

"Several of us have had conversations with President Trump about not weighing in on the race. And this is me: I don't think it's worth the president using the political capital that he has to weigh in on the race," said Senator Markwayne Mullin, a Thune supporter.

Thune, who is currently McConnell's top lieutenant, and Cornyn, who held the same job earlier, are veteran legislators who have helped usher major legislation through Congress and aided the campaigns of fellow Senate incumbents and other Republican candidates.

Scott is still serving his first six-year term in the Senate.

Asked about the pressure campaign, Thune said, "We're just keeping our heads down, doing the work and leaving it in the hands of the voters."

Scott, who served eight years as governor of Florida before he was elected to the Senate, said he was appreciative of his supporters. "I assume they've watched what I've done over the last 14 years in my job as governor and my job up here, and they think I would do a good job," he added.

Cornyn declined to answer questions about the pressure campaign.

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Reporting by David Morgan; Editing by Scott Malone and Bill Berkrot