Lawyers for former President Donald J. Trump said on Thursday that they want the special counsel, Jack Smith, and two of his top deputies to be held in contempt of court and sanctioned for violating a judge’s order that effectively froze the criminal case accusing Mr. Trump of plotting to overturn the 2020 election.

The lawyers in their request seek to force Mr. Smith and his team to explain why they should not be held in contempt and possibly pay a portion of Mr. Trump’s legal fees. The request was the latest aggressive move in what has quickly turned into a legal slugfest between the defense and prosecution, underscoring how critical the issue of timing has become in the election subversion case.

The spat began last month when Judge Tanya S. Chutkan, who is overseeing the case in Federal District Court in Washington, put all of its proceedings on hold until Mr. Trump resolved his attempts to have the underlying charges dismissed with claims that he has immunity from prosecution in the case.

Those arguments will be heard on Tuesday by a federal appeals court in Washington and are likely to make their way to the Supreme Court for another level of review.

The trial in the election case is set to begin in early March. Hoping to keep it on schedule, prosecutors working for Mr. Smith have, on occasion, sought to nudge the matter forward despite Judge Chutkan’s order.

A few days after the order was imposed, for instance, they told the judge that they had sent Mr. Trump’s legal team a draft list of exhibits that they intended to use at the trial and thousands of pages of additional discovery materials. They noted that the list and the documents had been turned over “to help ensure that trial proceeds promptly if and when” the case was back in action.

Then, two days after Christmas, the prosecutors filed a memo to Judge Chutkan, asking her to stop Mr. Trump from making “baseless political claims” or introducing “irrelevant disinformation” at the trial.

After Mr. Smith sent the draft list of exhibits, lawyers for Mr. Trump fired off an angry letter to Judge Chutkan, complaining about how prosecutors had “improperly and unlawfully attempted to advance this case” in violation of her order pausing it.

But the lawyers were silent about Mr. Smith’s second such move until Thursday.

In a 15-page motion, John F. Lauro, writing for Mr. Trump’s legal team, accused the prosecution of “partisan-driven misconduct” and said they had treated Judge Chutkan’s decision to pause the case as “merely a suggestion meaning less than the paper it is written on.”

Mr. Lauro also asked for a series of potentially severe consequences, starting with an order that would force Mr. Smith and two of his deputies — Thomas P. Windom and Molly Gaston — to come up with answers for why they should not be held in contempt and be made to pay whatever legal fees Mr. Trump may have incurred by dealing with their recent filings and productions.

Moreover, Mr. Lauro asked the judge to make the prosecutors tell her why they should not be forced to “immediately withdraw” the last motion they filed and be “forbidden from submitting any further filing” without express permission.

“These were no accidents,” Mr. Lauro wrote about Mr. Smith’s attempts to keep pushing the case forward. “The submissions were fully planned, intentional violations of the stay order, which the prosecutors freely admit they perpetrated in hopes of unlawfully advancing this case.”

The skirmish over the stay order reflects how central the question of timing is to the election interference case. In addition to the back and forth about legal issues large and small, the defense and prosecution have been waging a second war over when the case will go to trial — specifically, if it will be held before or after the 2024 election.

For weeks, Mr. Smith and his team have been trying to keep the trial on schedule, arguing that the public has an enormous interest in a speedy prosecution of Mr. Trump, who is the Republican Party’s leading candidate for the presidency. In doing so, they have gone to unusual lengths, at one point making a failed request to the Supreme Court to leap ahead of the appeals court that is now hearing Mr. Trump’s immunity claims and to render a quick decision.

Mr. Trump’s lawyers have used every means at their disposal to slow the case down, hoping to delay a trial until after the election is decided. If that happened and Mr. Trump won, he would have the power to order the federal charges against him dropped.