Mr. Cohen’s goal for the meeting — and whether it may have been related to his consulting business — remains unclear. His lawyers, and lawyers for Mr. Vekselberg, did not respond to requests for comment when told about video footage from C-Span showing Mr. Vekselberg and Mr. Intrater arriving at Trump Tower on Jan. 9, 2017. Mr. Intrater said the meeting included only a brief discussion about relations between the United States and Russia.

The meetings and Columbus Nova’s payments to Mr. Cohen have attracted scrutiny from Robert S. Mueller III, the special counsel investigating Russian interference in the election, as well as federal prosecutors in Manhattan examining Mr. Cohen’s business activities and finances, the people said.

Early this year, Mr. Mueller’s investigators stopped Mr. Vekselberg at a New York-area airport after he arrived on a private plane, The New York Times reported in early May. Mr. Mueller’s investigators have interviewed Mr. Intrater twice, focusing partly on his dealings with Mr. Cohen. Mr. Mueller’s office has since referred the investigation of Mr. Cohen to the prosecutors in Manhattan. There is no indication that Mr. Vekselberg or Mr. Intrater is suspected of wrongdoing, and Mr. Intrater’s lawyer, Richard Owens, said that “Columbus Nova has cooperated with all requests for documents and information from federal authorities.”

The meetings were an opportunity for the three men to discuss shared interests during the presidential transition and early days of the administration, according to interviews and records reviewed by The Times.

Image Andrew Intrater, Mr. Vekselberg’s cousin, leads the firm that awarded the contract. Credit... Duane Prokop/Getty Images for the USC Shoah Foundation

Mr. Cohen, who promoted his connection to Mr. Trump, was seeking moneymaking opportunities. Mr. Vekselberg, who has longstanding business and philanthropic ties to the United States and controls a global conglomerate, was interested in Mr. Trump’s Russia-friendly stance. And Mr. Intrater, looking for new investors, was drawn to Mr. Cohen’s Rolodex of rich donors — he is a deputy finance chairman of the Republican National Committee — and the lure of new deals to invest in, prompting Columbus Nova to hire him as a consultant.

By all accounts, it did not work out as planned for any of the men. Mr. Cohen is under investigation; the Trump administration hit Mr. Vekselberg with sanctions last month, making him one of seven Kremlin-linked oligarchs to be punished as part of a response to meddling and other aggressions; and Columbus Nova parted ways with Mr. Cohen after no new deals or investors materialized, despite paying him $580,000 of the $1 million contract.