The discussion took place in September 2016, in the lead-up to the presidential election. "The recording. between Trump and Cohen. confirms the man who now occupies the Oval Office had contemporaneous knowledge of a proposal to buy the rights to the story of Karen McDougal, a woman who has alleged she had an extramarital affair with Trump", Stris tweeted. Trump's advisers are also considering whether they should instead play along and release damning recordings of Cohen, according to two people close to the President.
In the at-times muddled recording, Mr Cohen says he wants to set up a company to acquire the rights off American Media, the company that owns the Enquirer.
"Pay with cash", Trump appears to respond. Cohen counters by repeatedly saying, "No". The suit says the foundation routinely and willfully broke the law by engaging in self-dealing that ultimately came to include illegal coordination with Trump's political campaign. Davis is a columnist for The Hill.
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. While Davis said Trump was suggesting the two pay cash, Giuliani's version of the transcript says Trump is saying, "Don't pay with cash...check".
Meanwhile, a source told CNBC that waiving the privilege claim over that tape provided Giuliani the ability to release "his version of the tape's contents". Davis does acknowledge that Trump used the word "check" toward the end of the tape. "After Michael Cohen mentioned financing".
The tape doesn't prove McDougal's allegations of an affair and at no point in the recording does Trump acknowledge whether he had a sexual relationship with McDougal.
Trump then turns his attention to Cohen, his longtime friend and attorney.
Aired exclusively by CNN Tuesday night, Cohen's tape appears to contradict repeated claims by the president's team that Trump had no knowledge of any payments to silence McDougal. That may explain why prosecutors did not follow typical Justice Department protocol to avoid subpoenaing news organizations when possible, and to give journalists advance warning when demanding documents or other information.
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Davis told CNN he released the tape to dispute an assertion by Giuliani that the tape would show that Trump made clear that if there were going to be a payment, it should be done by cheque, which would be easily traced.
"This tape is more evidence that Donald Trump himself was directly involved in a campaign finance law violation", Rep. Ted Lieu (D-Calif.) declared on Twitter. White House spokeswoman Sarah Sanders said this week "the president maintains he's done nothing wrong".
During the conversation, Cohen assured Cuomo he was not running a tape, the Journal reported - telling the reporter he was putting the phone on his desk and the conversation was off the record.
The existence of the tape, secretly recorded by his former longtime attorney Michael Cohen two months before the 2016 election, was reported last week by the New York Times, which said the Federal Bureau of Investigation seized it during a raid on Cohen's office this year.
Mr Trump's current lawyer, Mr Rudy Giuliani, said he got an audio expert to enhance the recording, the Associated Press reported.
Media captionWas Trump's Stormy Daniels payment legal?
Mr Cohen says: "We'll have to pay".