This comes after CBS News reported Cohen taped then-candidate Trump discussing a possible payment to Playboy model Karen McDougal, who claimed she once had a year-long affair with the future president.
The prosecutors working on the case had not reviewed the recording because it, along with millions of other documents and files seized in Federal Bureau of Investigation searches of Cohen's home, hotel room and office in April, was undergoing the special master process, in which an independent party reviews whether the items should be regarded as privileged and thus withheld from prosecutors.
He said the FBI's raid on Cohen's office was "inconceivable", but alleged Cohen's actions were "even more inconceivable" and "perhaps illegal". Cohen has not been charged with a crime.
After the initial raid on Cohen's office and hotel room, Trump unleashed a furious assault on Special Counsel Robert Mueller, saying of the raid, "It's an attack on our country. what we all stand for".
Lanny Davis, an attorney for Cohen, said, "Obviously, there is an ongoing investigation, and we are sensitive to that".
McDougal has said she began a almost year-long affair with Trump in 2006 shortly after his wife, Melania, gave birth.
The matter 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 USA campaign finance laws. Maybe the rest of the Cohen-Trump tapes are just the two of them talking about the dismal state of the Knicks.
Cohen accidentally recorded the conversation with trump two months before the U.S. presidential elections in 2016. Presumably Trump and Cohen wanted to take over the rights to the tape so that they could be sure about the scandal staying on the down low.
Ms McDougal sold her story to The National Enquirer for $150,000 during the final months of the presidential campaign but the tabloid sat on the story, preventing it from becoming public, the Times said.
The development was first reported Friday by The New York Times.
The chairman of its parent company, American Media, is a friend of Trump's. Trump's campaign had said it knew nothing about any payment to McDougal, the former Playboy centerfold. "When all of this evidence ultimately comes to light, and I hope that it will. this is going to spell a significant problem for Michael Cohen and a larger problem for the president".
Mr Trump was said to have only found out earlier this week that Mr Cohen had recorded him.
If it came from the Cohen side, that's yet more evidence that the man who once said he would take a bullet for Trump may be softening in his commitment to his former boss amid the possibility of significant jail time.
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