Manhattan prosecutors on Monday asked the judge overseeing the criminal case against Donald J. Trump to prohibit the former president from attacking witnesses or exposing jurors’ identities.

The requests, made in filings by the Manhattan district attorney’s office, noted Mr. Trump’s “longstanding history of attacking witnesses, investigators, prosecutors, judges, and others involved in legal proceedings against him.”

In outlining a narrowly crafted gag order, the office hewed closely to the terms of a similar order upheld by a federal appeals court in Washington in another of Mr. Trump’s criminal cases.

The gag order in the Manhattan case, if the judge approves it, would bar Mr. Trump from “making or directing others to make” statements about witnesses concerning their role in the case. The district attorney, Alvin L. Bragg, also asked that Mr. Trump be barred from commenting on prosecutors on the case — other than Mr. Bragg himself — as well as court staff members.

Although Mr. Bragg carved himself out of the gag order request, the district attorney has received the brunt of the attacks from Mr. Trump and his supporters. In an affidavit released Monday, the head of his security detail listed some of the worst of the dozens of attacks directed at Mr. Bragg last year, including racial slurs and death threats.

In a separate filing, Mr. Bragg placed a special emphasis on the protection of jurors. His prosecutors asked that Mr. Trump be barred from publicly revealing their identities. And although Mr. Trump and his legal team are allowed to know the jurors’ names, Mr. Bragg asked that their addresses be kept secret from the former president.

If the judge, Juan M. Merchan, accepts the restrictions, he would be just the latest judge to impose a gag order on the former president. There was an order in the Washington case, a federal case that involves accusations that Mr. Trump plotted to overturn the 2020 election. And the judge in Mr. Trump’s civil fraud trial that recently concluded ordered Mr. Trump not to comment on court staff members.

In a federal trial in Florida in which Mr. Trump is accused of mishandling classified documents, the special counsel, Jack Smith, is also seeking to protect witnesses. The prosecutors have vehemently opposed an attempt by Mr. Trump’s lawyers to include the names of about 24 potential witnesses in a public filing, claiming that the witnesses could face harassment or intimidation. The prosecutors have even opened a separate criminal investigation of threats made on social media against one of the witnesses.

The Manhattan criminal case was the first of Mr. Trump’s four indictments to be filed and is scheduled to go to trial on March 25. Last year, the district attorney’s office accused Mr. Trump of 34 felonies, saying he had orchestrated a cover-up of a potential sex scandal with a porn star that could have hindered his 2016 presidential campaign.

Mr. Trump’s lawyers will be likely to oppose the gag order and could appeal it if Justice Merchan adopts it. A lawyer for Mr. Trump, Todd Blanche, declined to comment on the prosecutors’ proposal, saying that the defense’s court papers spoke for themselves.

Steven Cheung, a spokesman for Mr. Trump’s campaign, slammed Mr. Bragg’s request for a “restrictive gag order, which if granted, would impose an unconstitutional infringement on President Trump’s First Amendment rights, including his ability to defend himself, and the rights of all Americans to hear from President Trump.”

The former president has reveled in public attacks on his former fixer, Michael D. Cohen, who is now one of Mr. Bragg’s key witnesses. Mr. Cohen paid $130,000 in hush money to the porn star to silence her story of an affair with Mr. Trump and was later reimbursed by Mr. Trump.

The case hinges on the reimbursement to Mr. Cohen. Mr. Trump, prosecutors say, knew that his company had falsified internal records, referring to the reimbursement as “legal expenses” that were part of a “retainer agreement.”

Mr. Bragg has cast Mr. Trump’s actions as election interference, arguing that the cover-up led to the withholding of important information from voters shortly before they headed to the polls.

But in their own filing Monday, Mr. Trump’s lawyers asked that the prosecutors be barred from asserting that their client had sought to influence the election, saying the argument was irrelevant.

Mr. Bragg, the defense wrote, was saying that “efforts by a candidate to prevent adverse publicity about himself during a campaign equals an attempt to defraud.” They said the argument was not based on the law and “is an extraordinary perversion of our election system and the First Amendment.”

Mr. Trump’s defense team also asked that the judge prevent Mr. Cohen from testifying.

“Michael Cohen is a liar,” the former president’s lawyers wrote, accusing Mr. Cohen of perjury in Mr. Trump’s civil fraud trial and saying that his public statements indicated that he planned to lie again. (The judge in the civil fraud case concluded that Mr. Cohen had “told the truth.”)

Mr. Cohen himself fired back on Monday, saying in a text message, “As the March 25th date draws closer and closer, Donald and his legal team of misfits will attempt to concoct new ways to delay this case.”

In a separate motion filed on Monday, prosecutors provided something of a guide to their case, signaling that they hope to include other hush-money payoffs they say Mr. Trump orchestrated: one with a former Playboy model, and another involving a doorman who sought to sell an embarrassing story about Mr. Trump in 2015.

As they laid the groundwork to tell that expansive story, they asked Justice Merchan to allow them to introduce evidence from the 2016 presidential campaign, including the infamous “Access Hollywood” recording in which Mr. Trump boasts about groping women, as well as three public allegations of sexual assault made against him after the recording was released.

It is far from clear that Justice Merchan will grant those requests. Persuading him that allegations of sexual assault should be allowed could be particularly difficult, given that judges are supposed to carefully evaluate evidence that could unfairly harm a defendant in the eyes of the jury.

Prosecutors on Monday were more worried about the defendant harming jurors. In seeking to limit Mr. Trump from disclosing their names, the district attorney’s office cited recent instances of his attacking jury members, including in the 2020 trial of his associate Roger Stone. While still president, prosecutors noted, Mr. Trump had called the head of that jury “totally biased,” “tainted” and “DISGRACEFUL!”

Alan Feuer contributed reporting.