Federal prosecutors moved on Wednesday to drop their last remaining efforts to prosecute President Trump’s two co-defendants in the classified documents case brought by the former special counsel Jack Smith.

In a single-page filing, prosecutors asked a federal appeals court in Atlanta to dismiss an appeal that had been filed before Mr. Trump took office seeking to reinstate criminal charges against the two men charged with Mr. Trump, Walt Nauta and Carlos De Oliveira.

If the appeals court agrees to drop the government’s challenge, it would be the formal end of the case in which Mr. Trump was accused of illegally holding onto classified materials after he left office in 2021 and then of conspiring with Mr. Nauta and Mr. De Oliveira to obstruct the government’s repeated efforts to retrieve them.

The case ended up in front of the appeals court this summer after Judge Aileen M. Cannon, who was appointed by Mr. Trump, dismissed the charges in their entirety. Her ruling determined — against decades of legal precedent — that Mr. Smith had been unlawfully appointed to his post as special counsel.

While Mr. Smith’s deputies challenged that decision, they were ultimately forced to drop the appeal where Mr. Trump was concerned after he was re-elected in November. A binding Justice Department policy prohibits pursuing criminal prosecutions of a sitting president.

But prosecutors did not immediately seek to drop their appeal regarding Mr. Nauta and Mr. De Oliveira. The case remained in front of the U.S. Court of Appeals for the 11th Circuit after Mr. Trump took office and the top leadership of the Justice Department was replaced by his appointees.