The newspaper, citing lawyers and others familiar with the recording, said the FBI seized the recording this year during a raid on Cohen's office.
Federal prosecutors in New York are investigating Cohen for possible bank and tax fraud, and for possible campaign law violations linked to a $130,000 payment to adult film actress Stormy Daniels and other matters related to Trump's campaign, a person familiar with the investigation has told Reuters.
The existence of a recording raises further questions about the tactics Trump used to keep his personal and business life secret while he was seeking public office.
Representatives for Cohen and Trump's attorney Rudy Giuliani did not immediately respond to requests for comment from Reuters.
Giuliani told the Times that Trump had discussed the payments with Cohen on the tape but that the payments were ultimately never made. He said the recording was less than two minutes and demonstrated the president had done nothing wrong.
The former model, Karen McDougal, said she began a nearly yearlong affair with Trump in 2006 shortly after his wife, Melania, gave birth to their son, Barron, the Times said.
McDougal sold her story to The National Enquirer for $150,000 during the final months of the presidential campaign but the tabloid sat on the story, preventing it from becoming public, the Times said.
Cohen's lawyer, Lanny Davis, declined to comment on the tapes, the newspaper said.
The newspaper, citing three people briefed on the matter, said Cohen's lawyers discovered the recording during their review of the seized materials and shared it with Trump's lawyers.