Mr. Manafort has told associates that he does not believe that Mr. Kilimnik has ties to Russian intelligence, but the document released on Tuesday shows that Mr. Gates told others of his history in the intelligence services. That history was widely discussed for years among people who worked with Mr. Manafort and Mr. Gates in Ukraine.

At the time of the calls, Mr. Gates was the Trump campaign’s liaison to the Republican National Committee and, before that, he was the campaign’s deputy chairman. Mr. Manafort served as the campaign chairman until August 2016, when he resigned amid the growing controversy about his work in Ukraine.

Both Mr. Manafort and Mr. Gates were indicted last year for money laundering and other financial crimes committed while, the charges said, they tried to hide the money they received for their Ukraine work. Last month, Mr. Gates pleaded guilty to financial fraud and lying to investigators and has agreed to cooperate with Mr. Mueller’s investigation.

Mr. Manafort has vowed to fight the charges. In February 2017, he told The New York Times he had “never knowingly spoken to Russian intelligence officers, and I have never been involved with anything to do with the Russian government of the Putin administration or any other issues under investigation today.”