Eleven days before the presidential inauguration last year, a billionaire Russian businessman with ties to the Kremlin visited Trump Tower in Manhattan to meet with Donald J. Trump’s personal lawyer and fixer, Michael D. Cohen, according to video footage and another person who attended the meeting.

In Mr. Cohen’s office on the 26th floor, he and the oligarch, Viktor Vekselberg, discussed a mutual desire to strengthen Russia’s relations with the United States under President Trump, according to Andrew Intrater, an American businessman who attended the meeting and invests money for Mr. Vekselberg.

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The men also arranged to see one another during the inauguration festivities, the second of their three meetings, Mr. Intrater said.

Days after the inauguration, Mr. Intrater’s private equity firm, Columbus Nova, awarded Mr. Cohen a $1 million consulting contract, a deal that has drawn the attention of federal authorities investigating Mr. Cohen, according to people briefed on the inquiry.