It was unclear what legal recourse, if any, Mr. Parnas’s lawyers would have if Mr. Barr refused their request, seemingly a legal long shot.

The request comes as Mr. Parnas has catapulted to the center of Mr. Trump’s impeachment proceedings.

Over the last week, House Democrats have released a trove of Mr. Parnas’s text messages with Mr. Giuliani and other players in the Ukrainian pressure campaign. The records, which Mr. Parnas shared with the House Intelligence Committee, showed the extent of his involvement in the campaign, contradicting Mr. Trump’s allies who have portrayed Mr. Parnas as a peripheral player, and spurring calls for him to testify at the Senate impeachment trial.

Mr. Parnas also has gone on something of a publicity tour, granting interviews to the MSNBC host Rachel Maddow and The New York Times and taking aim at the president. In his interview with The Times, he said, “I am betting my whole life that Trump knew exactly everything that was going on that Rudy Giuliani was doing in Ukraine.”

Mr. Trump has denied knowing Mr. Parnas, and the White House has dismissed his comments as self-interested lies told by a criminal defendant.

The lawyers also said in the letter that federal prosecutors in Manhattan had refused to meet with Mr. Parnas “and receive his information” regarding Mr. Trump, Mr. Giuliani and others, “all of which would potentially benefit Mr. Parnas if he were ever to be convicted and sentenced in his criminal case.”

Mr. Bondy told The Times last week that his client was seeking to cooperate with the federal prosecutors in Manhattan. “We very much want to be heard in the Southern District,” Mr. Bondy said. “We very much want to provide substantial assistance to the government.”

A formal cooperation deal, if granted by the government, can lead to a defendant’s receiving leniency when he is sentenced.