"He always told me he loved me," McDougal told CNN earlier this year. She told Anderson Cooper that she believed National Enquirer's purchasing of the story was a favor for Mr. Trump in the last few months of the presidential election.
The White House on Friday did not respond to CBS' inquiries, but has repeatedly denied the affair, issuing a statement in February saying "this is an old story that is just more fake news. The president says he never had a relationship with McDougal."
Cohen's attorney Lanny Davis said in a statement "when the recording is heard, it will not hurt Mr. Cohen. Any attempt at spin can not change what is on the tape."
The president's attorney, Rudy Giuliani, told The New York Times the recording also would not hurt his client. The New York Times first reported the existence of the tape of Cohen and Trump.
Payments to women have become a topic the White House and Mr. Trump's lawyers have had to address in recent months. In May, Giuliani revealed on Fox News that Mr. Trump repaid Cohen for a $130,000 payment Cohen made to adult film star Stormy Daniels ahead of the election in 2016. Daniels, like McDougal, claims to have had an affair with Trump. Giuliani insisted that there was no campaign finance violation in that instance.