Davis said the tape is among audio recordings that were seized by the Federal Bureau of Investigation in a raid of Cohen's home and office in April.
He added that he doesn't believe that Americans are angry at Trump, but rather are "embarrassed" and "sad" over the state of the country.
The conversation took place after The Journal in January revealed that Cohen had arranged the payment to Daniels.
The attorney for Michael Cohen, Donald Trump's longtime lawyer, says Rudy Giuliani is "inventing words" as he defends Trump after the release of a tape Cohen secretly made of Trump discussing a potential payment for a former Playboy model's account of having an affair with him. In the tape, Trump suggests paying "cash" for the rights to the story.
The approximately 2 minute recording, aired by CNN, has Cohen telling Trump that he'll have to set up "financing".
American Media purchased the story's rights from McDougal for $150,000 in August 2016.
"I need to open up a company for the transfer of all that info regarding our friend David", Cohen says.
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Will told O'Donnell that this first tape, in which Trump and Cohen discuss a payment to former Playboy model Karen McDougal, is just "redundant evidence" that Trump is "a seedy man whose incontinent sexual appetites get him into trouble with seedy women".
The paper reported that Cohen told Cuomo he wasn't running a tape - before recording their entire conversation.
Common Cause says the tape bolstered their argument that Trump knew about the payment and that it was potentially illegal. "Michael Cohen and his legal team were reluctant to respond out of respect for the process", the person said of the period after the existence of the tape became public.
Two months before the 2016 election, Donald Trump and his lawyer discussed plans for hush money payments to a former Playboy model alleging an affair. They also talk about mundane matters with the campaign, such as polling numbers. When Cohen says, "Well, I have to pay -", Trump interrupts and says, "Pay with cash".
Fox News chief judicial analyst Andrew Napolitano on Wednesday weighed in on a newly released audio tape of a conversation between President Donald Trump and his former personal attorney Michael Cohen, explaining that the conversation could constitute proof of fraudulent behavior and jeopardize Trump's attorney-client privilege.
Why did Lanny Davis release the tape?
There has been no comment from President Trump at this time.