A federal judge in Florida is holding a hearing on Friday to pick a new date for former President Donald J. Trump’s trial on charges of mishandling classified documents, a move that is likely to have major consequences for his legal and political future.

The judge, Aileen M. Cannon, has previously said she is inclined to make some “reasonable adjustments” to the timing of the trial, which for the moment, at least, is set to start on May 20 in Federal District Court in Fort Pierce, Fla. Several decisions Judge Cannon has reached in recent months about the pacing of the case have made it all but impossible for the trial to start as scheduled.

What remains to be seen is just how long of a delay Judge Cannon ends up imposing.

On Thursday evening, Mr. Trump’s lawyers and prosecutors in the office of the special counsel, Jack Smith, sent Judge Cannon their proposals about when the trial should begin.

Mr. Smith’s legal team, hewing to its long-held position of trying to conduct the trial before Election Day, requested a date of July 8. But after months of seeking to delay the trial until next year, Mr. Trump’s lawyers suddenly reversed themselves and suggested a date of Aug. 12.

The hearing in front of Judge Cannon, who was appointed to the bench by Mr. Trump in his waning days in office, is being held just days after a decision by the Supreme Court that increased the possibility that the former president might not face trial before Election Day in his other federal case — the one in which he stands accused of plotting to overturn the 2020 election.

The justices agreed to decide whether Mr. Trump is immune from prosecution on the election interference charges, scheduling arguments for the end of April and keeping the proceedings in the trial court frozen until they resolve the issue. As a practical matter, the Supreme Court’s decision to take up the case meant that the election trial was unlikely to begin before September, in the heat of the general election campaign.

Judge Cannon’s decision about whether to go with a July date, an August date or something later in the documents case could have an effect on the timing of the election case, as well. Mr. Trump attended the hearing on Friday.

It was not clear why Mr. Trump’s legal team said it would be open to August after long seeking to postpone the trial until next year. But one possibility was that the lawyers, by proposing to spend much of late summer and early fall in court on the classified documents case, were seeking to reduce the chances of there being time for the election case to go to trial before Election Day.

Only months ago, it seemed that Mr. Trump would spend much of 2024 in front of a jury, fending off four separate criminal indictments in four different cities.

At this point, however, only one of his criminal trials has a solid start date. Last month, a state judge in Manhattan picked March 25 for commencing his trial on charges of arranging hush-money payments to a porn star in an effort to avert a scandal on the eve of the 2016 election.

Mr. Trump’s fourth criminal case, in which he stands accused of tampering with the election results in Georgia, has not yet been set for trial. It is currently in turmoil as a judge in Fulton County considers whether to disqualify Fani T. Willis, the district attorney who filed the indictment, from the case over allegations of financial misconduct surrounding a romantic relationship she had with one of her deputies.

The hearing in Florida, which is scheduled to last most of Friday, will touch on more than scheduling issues.

Judge Cannon has asked the defense and prosecution to be ready to discuss Mr. Trump’s unusually broad and highly politicized motion for additional discovery, which was filed in January. In the motion, the former president’s lawyers suggested that, as part of their defense at trial, they intended to argue that federal officials — chief among them those from the intelligence community — were “politically motivated and biased” against Mr. Trump.

The parties are also set to debate an effort by Mr. Smith to keep under seal the names of about two dozen potential witnesses who could testify at trial.

Judge Cannon briefly agreed to a request by Mr. Trump’s lawyers to include the witnesses’ names in a public court filing. But she put that decision on hold after Mr. Smith accused her of having made a “clear error” and said the witnesses could face threats or harassment if their identities were revealed.

Even amid discussion of these other issues, the question of the trial’s timing was arguably paramount.

If Judge Cannon were to postpone the proceeding into next year, she would probably face a tidal wave of criticism. It is possible she could also provoke Mr. Smith’s first appeal since the indictment was returned in June, even though rulings related to scheduling matters are generally not subject to a challenge in higher courts.

When Judge Cannon was randomly assigned to the case last spring, she was already under fire for having issued a ruling in an early part of the inquiry that was favorable to Mr. Trump, but so legally questionable that an appeals court sternly rebuked her in reversing it.

After the F.B.I. searched Mar-a-Lago, Mr. Trump’s private club and residence in Florida, for classified documents in August 2022, Judge Cannon appointed an independent arbiter to figure out whether any of the materials collected by the agents were privileged and should be kept out of the hands of investigators.

But she accompanied that relatively typical decision with another that was all but unheard-of, effectively freezing the government’s investigation of Mr. Trump in place until after the arbiter, known as a special master, completed his work.

Prosecutors were outraged by the move, accusing Judge Cannon not only of lacking the power to insert herself into the case so extremely, but also of treating Mr. Trump differently than a normal criminal defendant.

A federal appeals court in Atlanta ultimately agreed, unanimously reversing her decision and pointing out that she appeared to have granted “a special exception” for Mr. Trump in defiance of “our nation’s foundational principle that our law applies to all.”

Still, in some of her more recent rulings, Judge Cannon has shown herself willing to buck Mr. Trump.

On Wednesday, for example, she denied a highly unusual request from his lawyers to gain access to a secret government filing detailing a trove of classified discovery evidence that prosecutors said was neither helpful nor relevant to his defense.

If Judge Cannon had permitted the request, legal experts said, it would have fallen far outside the normal procedures laid out in the Classified Procedures Act, the federal law governing the use of classified materials at public trials.

But even while ruling against Mr. Trump, Judge Cannon seemed to suggest that he was different from most criminal defendants. She did not quite agree with Mr. Smith’s position that the facts in this case did not “remotely justify a deviation from the normal process.”

“The court,” she wrote, “cannot speak with such confidence in this first-ever criminal prosecution of a former United States president — once the country’s chief classification authority over many of the documents the special counsel now seeks to withhold from him.”