The tapes were reportedly discovered during an FBI raid on Mr Cohen’s property earlier this year in New York.
The New York Times reports that on the recording Mr Trump and Mr Cohen discuss paying Karen McDougal, who says she had an affair with Mr Trump.
The tape was reportedly made two months before the election.
The Department of Justice is looking into alleged hush money paid to women who claim they had a relationship with Mr Trump.
Mr Cohen – who has not been charged – is reportedly under investigation for possible bank and tax fraud, as well as potential violation of election law.
On Friday, another Trump lawyer, Rudy Giuliani, confirmed to the New York Times that the president had discussed the payments with Mr Cohen on the tape.
But he said no money was actually ever transferred, and the recording itself amounted to “powerful exculpatory evidence”.
In the lead-up to the 2016 presidential election, Ms McDougal sold her story to the National Enquirer, which is owned by a personal friend of Mr Trump.
She says the $150,000 (£115,000) agreement gave the supermarket tabloid exclusive story rights and banned her from talking publicly about the alleged affair.
But the Enquirer did not publish her kiss-and-tell, and she says she was tricked.
The Washington Post reported that Mr Trump and Mr Cohen discussed possibly paying American Media Inc, the parent company of the National Enquirer, for the rights to Ms McDougal’s story.
Federal investigators have reportedly demanded the tabloid’s records on the McDougal payment.
Ms McDougal says she had a 10-month affair with Mr Trump in 2006, a year after he wed Melania Trump.
When questioned about the allegations, President Trump has denied the affair and said he had no knowledge of any payment.
In May, President Trump admitted that he had reimbursed Mr Cohen for a payment he made to another woman to hush up her claims of an affair.
Mr Trump had previously denied all knowledge of the $130,000 payment to porn star Stormy Daniels as part of a non-disclosure agreement.
Ms Daniels says she and Mr Trump had sex in a hotel room in Lake Tahoe, a resort area between California and Nevada, in 2006.
Non-disclosure agreements of the type signed by Ms Daniels and allegedly by Ms McDougal are not in themselves illegal.
It is a potential problem for Mr Trump because undisclosed payments to bury embarrassing stories about a political candidate can be treated as a violation of US campaign finance laws.
Mr Cohen once famously vowed he would take a bullet for Mr Trump.
But he told ABC News this month that his loyalty to his family and country came before his old boss.
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.
The president has said that Mr Cohen is no longer his personal lawyer.