Trump and Cohen have denied the affair.
The payments tied to Vekselberg came from a U.S. firm called Columbus Nova, which has financial ties to Renova, a Moscow-based global conglomerate run by Vekselberg. Columbus Nova said in a statement to ABC News late Tuesday that the company's work with Cohen was not improper.
"Columbus Nova is a management company solely owned and controlled by Americans. After the inauguration, the firm hired Michael Cohen as a business consultant regarding potential sources of capital and potential investments in real estate and other ventures,” Columbus Nova's attorney Richard Owens said in the statement. “Reports today that Viktor Vekselberg used Columbus Nova as a conduit for payments to Michael Cohen are false. The claim that Viktor Vekselberg was involved or provided any funding for Columbus Nova's engagement of Michael Cohen is patently untrue. Neither Viktor Vekselberg nor anyone else other than Columbus Nova's owners, were involved in the decision to hire Cohen or provided funding for his engagement."
Vekselberg, who is considered one of the richest men in Russia, reportedly holds frequent meetings with Putin as part of a business group known as the Russian Union of Industrialists and Entrepreneurs. ABC News' emails to Vekselberg’s company seeking comment were not returned.
The White House declined to comment Tuesday.
Avenatti's law firm released the seven-page document amid a federal investigation into Cohen. Federal agents last month searched Cohen's home, hotel room and office for documents and computer files, including those related to his $130,000 payment to Daniels in October 2016.
Cohen has not been charged with any crime, and both he and his attorney have maintained he has done nothing improper.
When asked on "GMA," Avenatti declined to say how he obtained the records but said he's "100 percent" confident they're accurate.
"We stand behind it, 100 percent there's no question it’s accurate," he said in the interview this morning. "We haven't released all the evidence that we have."
The attorney reiterated his calls to the president and Cohen to disclose "what money was received or not received."
"Something doesn’t smell right," Avenatti said. "If there's nothing to hide then why do they continue to hide this information, hide the involvement?"
ABC News' James Hill, Tom Llamas and Matthew Mosk contributed to this report.