Why Mr. Manafort wanted them to see American polling data is unclear. He might have hoped that any proof that he was managing a winning candidate would help him collect money he claimed to be owed for his work on behalf of the Ukrainian parties.

About the same time, Mr. Manafort was also trying to curry favor with Oleg V. Deripaska, a Russian billionaire close to the Kremlin and an associate of Mr. Kilimnik. In July 2016, Mr. Manafort, then the Trump campaign chairman, told Mr. Kilimnik that he could offer Mr. Deripaska “private briefings,” according to emails reported by The Washington Post. Mr. Deripaska had claimed Mr. Manafort owed him millions from a failed business venture, and Mr. Manafort may have been trying to use his status in the campaign to hold him at bay.

The surprise disclosures about Mr. Manafort were the latest in two years of steady revelations about contacts between associates of Mr. Trump’s and Russian officials or operatives. In another development on Tuesday, the Russian lawyer who met with senior campaign officials at Trump Tower in June 2016 was charged with obstruction of justice in an unrelated case. Federal prosecutors in Manhattan said that the lawyer, Natalia V. Veselnitskaya, had pretended to a federal judge that she was purely a private defense lawyer when in fact she was working with the Russian government to thwart the civil prosecution of a Russian company.

Of the various Russian intermediaries to the Trump campaign, Mr. Kilimnik appears to be one of the most important to Mr. Mueller’s inquiry. A Russian citizen and resident, he faces charges from Mr. Mueller’s team of tampering with witnesses who had information about Mr. Manafort, but Mr. Kilimnik is not expected to ever stand trial. He did not respond to a phone call and email seeking comment on Tuesday.

His relationship with Mr. Manafort dates back years. The two men worked together to promote a Russia-aligned politician, Viktor F. Yanukovych, who rose to Ukraine’s presidency, was ousted in a popular uprising and fled to Russia in 2014. The two men continued working together over the next three years as Mr. Manafort’s financial troubles grew and investigators began to investigate a fraud scheme that eventually led to his conviction for 10 felonies.

In August 2016, apparently just before Mr. Manafort was fired from the Trump campaign, he and Mr. Kilimnik met to discuss a plan for Ukraine that seemed to further Russia’s interests. They also met several times afterward, including once in Madrid in early 2017. In an interview in February 2017 in Kiev, Ukraine’s capital, Mr. Kilimnik suggested the plan would have involved reviving the political fortunes of Mr. Yanukovych, the ousted Ukrainian leader.

For Russia, trying to influence the incoming Trump administration’s policy on Ukraine was of paramount importance. The economic sanctions imposed after Russia annexed Crimea damaged the Russian economy, and various emissaries have tried to convince administration officials to broker a resolution to a long-running guerrilla war between Russia and Ukraine.