About one month before the recording was made, the rights to McDougal's story were bought by American Media Inc., the company that publishes the National Enquirer, The Wall Street Journal reported.
In the tape, Trump can be heard discussing the logistics of financing and whether to "pay with cash," although the sound is muffled and Trump's instructions on that are unclear. Trump's lawyer, Rudy Giuliani, told Fox News that the president actually said, "Don't pay with cash."
He repeated the claim Wednesday on Twitter.
The recording was one of several seized in April by federal authorities conducting search warrants on Cohen's hotel and office, according to The New York Times. Other items seized included Cohen's computer, his phone and several records, The Washington Post reported.
Authorities sought details on Cohen's efforts to stave off negative publicity of Trump, CBS News and the Times reported. Among other things, authorities sought information on the release of an infamous tape in which the president could be heard on a hot mic making derogatory comments about women and payments Cohen made to a pair of women who claim they had sexual relationships with Trump, including McDougal and adult film star Stormy Daniels, according to the Times.
Cohen's attorney has said he released the tape after being subjected to an "intense campaign of disparagement" by the president's team.
The Associated Press contributed to this report.