https://www.bbc.com/news/world-us-canada-66828011.amp
Special Counsel Jack Smith's filing says the "narrowly tailored" order would prevent harassment of witnesses.
Mr Trump hit back online, accusing Smith's team of misconduct, writing: "they won't allow me to SPEAK?"
He has pleaded not guilty to conspiracy to overturn the 2020 election.
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The request was unsealed by District Court Judge Tanya Chutkan and was filed a week earlier. It was among a slew of older court documents from the case that were released on Friday.
Prosecutors say their proposed order - which they never refer to as a "gag order" - is "a narrow, well defined restriction" that is necessary to prevent disinformation, threats and "prejudicing" the case.
If approved, it would ban Mr Trump from making statements "regarding the identity, testimony, or credibility of prospective witnesses" and "statements about any party, witness, attorney, court personnel, or potential jurors that are disparaging and inflammatory, or intimidating".
It does not place any restrictions on Mr Trump from quoting from public record court documents or proclaiming his innocence.