Federal prosecutors and lawyers for former President Donald J. Trump will square off on Monday over whether a gag order should be put on Mr. Trump to restrict his often threatening statements about his federal indictment on charges of conspiring to overturn the 2020 election.

The issue is one of the most contentious to have arisen in any of the criminal cases Mr. Trump is facing. It sets up a momentous battle between the First Amendment rights of a candidate for the country’s highest office and the prerogatives of a judge to protect witnesses in a case she is overseeing and to preserve the integrity of the proceedings against bullying threats and disruptions.

The prosecution and defense are scheduled to gather at 10 a.m. in Federal District Court in Washington. They will make their arguments both for and against the order to Judge Tanya S. Chutkan, who was assigned to the case when Mr. Trump was charged in August. It is possible Judge Chutkan could rule from the bench by the end of the hearing.

Gag orders limiting what trial participants can say outside of court are not uncommon. But Mr. Trump’s status as the front-runner for the Republican presidential nomination, and his decision to portray the gag order request as part of an effort by the Biden administration to stifle the voice of a political rival, makes this request by prosecutors especially fraught and complex.

Legal experts have said there are few precedents for how Judge Chutkan should think about how to weigh strong constitutional protections for political speech against ensuring the functioning of the judicial process and the safety of the people participating in it.

It also remains unclear how Judge Chutkan might choose to enforce a gag order should she end up imposing one and Mr. Trump violates it. The penalties could include fines or even jail time.

The prosecutors working for the special counsel, Jack Smith, first proposed putting a limited gag order on Mr. Trump last month, claiming that his “near-daily” social media attacks on people involved in the case were a menace to witnesses and threatened to taint the pool of jurors who would ultimately sit in judgment of the former president.

They cited a litany of verbal assaults that Mr. Trump had directed at players in the election case, including Mr. Smith (whom Mr. Trump has repeatedly called “deranged”) and Judge Chutkan herself. (The former president has amplified posts by others describing her as “a radical Obama hack.”)

Some of the former president’s more outrageous statements seem to have had real-world consequences. One day after he posted a message on his social media platform that read, “IF YOU GO AFTER ME, I’M COMING AFTER YOU!” a Texas woman left a voice mail message for Judge Chutkan, saying, “If Trump doesn’t get elected in 2024, we are coming to kill you, so tread lightly.”

The woman has since been arrested.

Mr. Trump’s lawyers have reacted with outrage to the gag order request, saying that any attempt to “muzzle” him during his presidential campaign would grossly violate his free speech rights.

“The prosecution would silence President Trump, amid a political campaign where his right to criticize the government is at its zenith, all to avoid a public rebuke of this prosecution,” one of the lawyers, Gregory M. Singer, wrote.

Mr. Trump was placed under a very limited gag order early this month by the New York State judge overseeing his civil trial in Manhattan, where he stands accused of inflating the value of his properties for years. That order restricts Mr. Trump from speaking about any people who work on the judge’s staff. The order Judge Chutkan is considering could be significantly broader.

Almost from the moment Mr. Trump was indicted, his legal team has raised a First Amendment defense, arguing that prosecutors had essentially charged him for voicing his opinions about the 2020 election.

Mr. Trump has since leaned into those claims, placing the election interference case — and the three other criminal indictments he is facing — at the heart of his campaign. His core political argument, that he is being persecuted, not prosecuted, may be protected in some ways by the First Amendment, but it has also put him on what could be a collision course with Judge Chutkan.