The New York Times reported the recording was made secretly by Mr Trump’s longstanding lawyer in 2016, two months before the election.
The recording has now been seized by the FBI in a raid on Mr Cohen’s office, as part of an investigation by the Department of Justice into whether federal campaign finance laws were breached.
The model in question is Karen McDougal, who claimed she had a ten-month affair with Mr Trump in 2006, shortly before the birth of Melania and Mr Trump’s son, Barron.
Ms McDougal went public with her allegations in a tearful interview with CNN in March this year, in which she claimed the president “always told me he loved me.”
Ms McDougal had sold her story to the National Enquirer during the election campaign for $150,000 – but the paper never published, and she has said she feels “tricked” by the publication.
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Mr Trump’s representatives have denied the affair, and moments ago Mr Trump Tweeted: “The good news is that your favourite President did nothing wrong!”
What did Trump say in the recording?
According to the Times, former New York mayor Rudy Giuliani confirmed the recording exists and is two minutes long.
In it, Mr Trump discussed potential payments to Ms McDougal.
Mr Giuliani, who has now been recruited as Mr Trump’s personal lawyer, has spoken to the Associated Press (AP) about the recording.
He said that while the conversation occurred, the payments never materialised and Trump did nothing wrong.
Mr Giuliani told AP: “The transaction that Michael is talking about on the tape never took place, but what’s important is: if it did take place, the president said it has to be done correctly and it has to be done by cheque.”
He said the conversation included Mr Trump and Mr Cohen discussing buying the rights to the story from the National Enquirer.
Mr Cohen, who has not been charged, is also reportedly under investigation for possible bank and tax fraud, as well as the potential violation of election law.
His lawyer, Lanny J Davis said in a statement on Friday: “When the recording is heard, it will not hurt Mr Cohen. Any attempt at spin cannot change what is on the tape.”
Why is this serious for Trump?
This isn’t the first allegation of hush money to keep a woman quiet: porn star Stormy Daniels has claimed the president paid her $130,000 to keep quiet about their affair.
Mr Trump admitted in May that he had made a payment via Mr Cohen to an unnamed woman to keep her quiet, but he vehemently denies the affair with Ms Daniels.
Non-disclosure agreements such as the one that Ms Daniels and Ms McDougal would have signed are not illegal.
However, there is a potential problem for Mr Trump: Undisclosed payments to bury embarrassing stories about political candidates can be treated as a violation of US campaign finance laws.
Mr Cohen, who once vowed he’d take a bullet for Mr Trump, has now provided perhaps the most damning evidence against him.
He said recently in apparent turn of allegiance: “I put my family and country first.”
His comments stoked speculation that he might co-operate with investigators against Mr Trump, although there has been no confirmation so far that he has done so.
Mr Trump has confirmed Mr Cohen is no longer his personal lawyer.
And it seems the President is prepared to go to battle, tweeting today that is is “inconceivable that a lawyer would tape a client – totally unheard of & perhaps illegal.”