Congressional Republicans want to require Senate Democrats running for president to recuse themselves from a possible impeachment trial against President Trump, saying the trial would mix impeachment with their political aspirations.

The five Democratic presidential hopefuls in the Senate would serve as jurors in Mr. Trump’s impeachment trial and have an inherent conflict of interest, said Rep. Jason Smith, a Missouri Republican pushing a resolution that calls on the Senate to amend its rules to require their recusal.

“No senator should participate in this trial when they are able to abuse their power in the U.S. Senate by attempting to remove their political opponent and help their own political ambitions,” Mr. Smith said.

His office pointed out that the chief justice of the Supreme Court and not the vice president, whose own political fortunes are directly tied to the outcome of such a trial, would oversee the proceedings as a presumably nonpartisan arbiter.

The resolution, which also applies to a president in their first term in office, has more than two dozen GOP co-sponsors including Rep. Mark Walker, vice chairman of the House Republican Conference.

But the House GOP’s non-binding resolution is unlikely to go anywhere in the Democratic-controlled House.

For Democrats looking to oust Mr. Trump, the resolution only complicates the already difficult impeachment math.

The House can impeach Mr. Trump with a simple majority vote. In the Senate, it takes a two-thirds majority to convict and remove a president from office. Democrats, who are in the minority in the Senate, need 20 Republican votes to convict Mr. Trump, which is all but impossible in the current political environment. Taking five Senate Democrats out of the mix would make it even less plausible.

Mr. Walker said the senators currently running for president have all but abandoned their day jobs unless they think the time in Washington would benefit their campaigns.

“Suddenly returning to Washington in pursuit of their ‘Spartacus moment’ is an embarrassment to our standards of judicial impartiality and a poor excuse for public service,” said Mr. Walker, North Carolina Republican.

The “Spartacus moment” refers to a remark Sen. Cory A. Booker, New Jersey Democrat, made. During the confirmation hearings for Supreme Court Justice Brett M. Kavanaugh, Mr. Booker compared his self-sacrifice on a dubious rules issue to the gladiator who led a failed slave uprising against the Roman Republic.

Mr. Booker, with Sens. Michael Bennet of Colorado, Amy Klobuchar of Minnesota, Bernard Sanders of Vermont and Elizabeth Warren of Massachusetts, makes up the group of five sitting senators campaigning for the Democratic presidential nomination.

Sen. Kamala D. Harris of California recently dropped out of the race. Sen. Kirsten E. Gillibrand of New York also sought the nomination but ended her campaign in August.

House Democrats are speeding toward a vote on articles of impeachment against Mr. Trump, potentially by the end of December.

They’re investigating whether the president improperly pressured Ukrainian President Volodymyr Zelensky to start investigations into former Vice President Joseph R. Biden and the 2016 U.S. presidential election.

A Senate trial in January could require all senators to be in Washington six days a week for weeks at a time instead of out on the campaign trail.

Some pundits have speculated that an impeachment trial could benefit candidates such as Mr. Biden and Mayor Pete Buttigieg of South Bend, Indiana, who will be free to campaign in the early voting states in January ahead of the Feb. 3 Iowa caucuses.

But the senators remaining in Washington for the high-profile trial would be the ones enjoying political advantages, said Democratic strategist Brad Bannon.

“The limelight’s going to be [in] Washington, D.C. — not Des Moines and Manchester,” Mr. Bannon said.

Senators in the race said they’re duty-bound to do whatever the impeachment process requires.

“We have to do what we have to do,” Mr. Bennet said on CNN recently. “And I hope we do [it] in a way that builds confidence in the American people, rather than creates more of a lack of confidence like they have had the last few years.”

Still, candidates such as Mr. Bennet and Ms. Warren have already said they have enough evidence for them to likely convict Mr. Trump.

“You know, some things are more important than politics. I took a constitutional oath, and that’s it. I’ll be there,” Ms. Warren said on MSNBC.

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