Mr. van der Zwaan’s lies involved a 2012 report prepared by Skadden and used to defend Mr. Yanukovych, then the Russia-aligned president of Ukraine, from international criticism over the prosecution and incarceration of one of his political rivals, former Prime Minister Yulia V. Tymoshenko. State Department officials criticized the report, which purported to be the result of the law firm’s independent research, as a misleading account of the actions of Mr. Yanukovych’s government.

Mr. Mueller’s team scrutinized the report as part of its examination of the business dealings of Mr. Manafort and Mr. Gates in Ukraine, where Mr. Manafort worked for about a decade as a political consultant before he joined the Trump campaign. Among other charges, both men have been accused of “using one of their offshore accounts to funnel $4 million to pay secretly” for the report. It is unclear how much, if any, of that money went to Skadden or other firms with which they worked.

The Ukrainian authorities had begun their own investigation into payments for the report after Mr. Yanukovych fled Ukraine for Russia amid a popular uprising in 2014.

Mr. van der Zwaan told the special counsel on Nov. 3 that he had not spoken to Mr. Gates since mid-August 2016, even though Mr. Gates had called him the following month instructing him to get in contact with an unidentified mutual acquaintance, court papers show.

He also told prosecutors that he had not talked to that same acquaintance since 2014, even though he called that person in September 2016 to discuss possible criminal charges against Mr. Manafort, a law firm and a former Ukrainian justice minister. The phone conversation, conducted in Russian, was important enough that Mr. van der Zwaan recorded it, according to court papers. Mr. van der Zwaan followed up by calling Mr. Gates, recording that conversation as well.

One of those conversations involved payments that were described as “the tip of the iceberg,” Andrew Weissmann, a prosecutor in Mr. Mueller’s office, said in court. He did not elaborate.

Mr. van der Zwaan also acknowledged that he lied when he told investigators that he had only a “passive role in the rollout of the report,” according to court papers. In fact, prosecutors said, he had discussed with Mr. Gates and others how to publicize the report to make it appear less damning to Mr. Yanukovych’s government, including describing any mishandling of Ms. Tymoshenko’s prosecution as no more than “procedural” errors. Part of the strategy, Mr. Weissmann said in court, was to give an advance copy of the report to The New York Times.

