Trump lawyer Rudy Giuliani told Reuters on Friday Cohen recorded a conversation with Trump in September, two months before the 2016 election, in which they discussed buying the rights to a story by a former Playboy model, Karen McDougal, who said she had an affair with Trump.
On his Twitter handle, Trump wrote; "Inconceivable that the government would break into a lawyer's office (early in the morning) - nearly unheard of". "Even more inconceivable that a lawyer would tape a client - totally unheard of & perhaps illegal".
On Friday, another Trump lawyer, Rudolph Giuliani, confirmed to the New York Times that the president had discussed the payments with Mr Cohen on the tape.
She claims that she started an affair with Trump in 2006, just after his wife, Melania, had their son, Barron.
Mr Cohen, who no longer represents Mr Trump, is being investigated over whether he violated campaign finance laws. NY state, for example, has a "one-party consent" law, which makes it a crime to record an in-person or telephone conversation unless one party participating in the conversation consents.
"Nothing in that conversation suggests that he had any knowledge of it in advance", Giuliani told ABC News.
Federal investigators have reportedly demanded the tabloid's records on the McDougal payment.
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In August 2016, AMI, the parent company of the National Enquirer, paid McDougal $150,000 for the right to her story but never published an article based on her account. But suffice it to say that when the recording is heard, it will not hurt Mr. Cohen.
Yet "it remains murky whether and when a NY lawyer can ethically tape without advance disclosure", Harris said. According to CNN, "McDougal signed a contract with American Media Inc., the company that owns the National Enquirer, which barred her from publicly speaking about the alleged affair before the 2016 election".
Cohen's lawyer, Lanny Davis, quickly fired back, hurling a tweet at the president, and his lawyer, Rudy Giuliani, in a show of worsening relations between the president and his former attorney.
Cohen, who served for a decade as a lawyer at the Trump Organization, was known to sometimes record conversations with associates, store them digitally and then replay them for colleagues, as The Washington Post reported in April.
Mr Trump was said to have only found out earlier this week that Mr Cohen had recorded him. "Any attempt at spin cannot change what is on the tape".
Team Cohen remains surprised that Trump's lawyers removed the protective claim of privilege, given their view that the tape is harmful to the President.
Prosecutors are apparently interested in payments he made on Trump's behalf to porn actress Stormy Daniels, and other business dealings related to Trump. Cohen has not yet been arrested or charged.
The White House has said that Mr Trump denies having an affair with Ms McDougal and that he has denied having sex with Stormy Daniels.