On Friday, another Trump lawyer, Rudy Giuliani, initially told the New York Times that the president and Mr Cohen had discussed payments to Ms McDougal on the tape.
Trump did not mention Cohen by name but appeared to be responding to reports that Cohen recorded a conversation with him in September 2016 in which they discussed possible payments to former Playboy model Karen McDougal - who claimed she had an affair with Trump in 2006.
The woman has claimed she had an affair with Trump.
"I think we need to bring this in-house", Cohen tells Trump in the recording, a source told The Post.
That payout seems to be a significant part of a federal investigation into Cohen that's been going on for months, as the hush money may be in violation of campaign finance laws. If Michael Cohen is a patriot, then ALL of the tapes should be released to the American people. Now.
The president has said that Mr Trump is no longer his personal lawyer. Most states, including NY, allow for recordings of phone conversations with only the consent of one party; other states require all parties to agree to a recording or have mixed laws on the matter.
Trump claims Michael Cohen's phone-taping is ’perhaps illegal’
This gives the FBI access to the recording as part of the USA attorney for the Southern District of NY investigation into Cohen. 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.
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. In addition, Trump's attorneys have not claimed the recording is a privileged attorney-client conversation, a person familiar with the investigation told The Post on Friday. Around that time, Trump was campaigning heavily in Pennsylvania and New Hampshire, two states that do require two-party consent. But The Washington Post reports the conversation took place in NY, which is a one-party consent state - meaning that, legally, only one party in a conversation need consent for a recording of that conversation to be legal.
"It's pretty clear from the tape, it's the first time Trump is finding out".
The president's aides, who spent the first part of the week frantically trying to figure out how to clean up the aftermath of the news conference, have had little to say about accusations by women that Trump's lawyers had paid them for their silence in the wake of extramarital affairs.
Cohen's recording adds to questions about whether Trump tried to quash damaging stories before the election.
On the tape, Trump can be heard urging Cohen to make sure he properly documents the agreement to buy the rights and urges him to use a check - rather than cash - to keep a record of the transaction, the person said.
"In the big scheme of things, it's powerful exculpatory evidence", he said. CNN's Keith Allen has also just reported that Trump's reaction when he was informed that at least one tape was in the FBI's possession, was, "I can't believe Michael would do this with me".