Giuliani told the Times that the recording made by Cohen lasts for less than two minutes, and demonstrated that Trump committed no wrongdoing.
Giuliani was quoted as saying that Cohen and Trump discussed Trump paying McDougal for her story, separately from the payment she received from The Enquirer, which would guarantee her silence in the future.
“Nothing in that conversation suggests that he had any knowledge of [The Enquirer’s payment] in advance” of the conversation with Cohen, Giuliani said, according to the report.
Giuliani said that Trump had told Cohen that if he were to pay McDougal he should do so with a check, rather than cash, so that there would be record of the payment.
“In the big scheme of things, it’s powerful exculpatory evidence,” Giuliani told the newspaper.
But a source familiar with the recording told CNBC on Friday night that, in fact, it was Trump who raised the idea of paying McDougal with cash.
The source also said that it was Cohen who told Trump they would use a check to pay her if they ended up making such a payment.
Giuliani, former New York City mayor, did not immediately respond to CNBC’s request for a comment.
The White House, which has previously denied that McDougal had an affair with Trump, referred questions to Giuliani.
Lanny Davis, one of Cohen’s new lawyers, did not immediately respond to CNBC’s request for comment on the Times article.
“We have nothing to say on this matter,” Davis told The Times.
Cohen is the subject of an ongoing criminal probe by federal prosecutors in New York City. He has not been charged in that case.
Prosecutors are eyeing his business dealings, in addition to a $130,000 payment that Cohen made to Daniels, whose real name is Stephanie Clifford, shortly before the election.
Daniels has said the payment was to keep her quiet about a sexual encounter she says she had with Trump in 2006, the same year in which McDougal says her own affair with Trump began.
The White House has denied Daniels had sex with Trump. But the president has admitted reimbursing Cohen for the hush money payment to Daniels.
— Additional reporting by CNBC’s Eamon Javers and