The Times, citing "lawyers and others familiar with the recording", disclosed that Cohen secretly recorded their conversation about the payment to the former model, Karen McDougal.
"Inconceivable that the government would break into a lawyer's office (early in the morning) - nearly unheard of", he posted at 8:10 am.
The tapes were reportedly discovered during an Federal Bureau of Investigation raid on Mr Cohen's property earlier this year in NY.
While, such tapes do not necessarily imply guilt on the part of Cohen or the President, Trump's reaction certainly doesn't sound like something coming from an individual with nothing to worry about. 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".
Cohen's lawyer, Lanny Davis, declined to comment to the Times.
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He added that Trump had directed Cohen that if he were to make any payment, write a check, versus sending cash, so it could be documented.
The FBI investigation stemmed in part from a referral by the US special counsel's office, which is looking into possible coordination during the election campaign between Trump's aides and Russian officials. On Twitter, he scrubbed mentions and photos of Trump from a profile that previously identified him as "Personal attorney to President Donald J. Trump". David Pecker, the chairman of parent company American Media Inc (AMI), is Trump's friend.
Giuliani said the discussion of payment did not mean McDougal's claim of an affair was true and characterized it as an attempt to resolve false allegations that were "personally damaging" to Trump.
The White House has said Trump denies having sex with Daniels.
The recording was first reported Friday by The New York Times. Cohen hasn't been charged with any wrongdoing, but is under criminal investigation for a slew of potential crimes related to his financial and business dealings, including possible violations of campaign finance laws over the payments he made to the former Playboy model and porn actress Stormy Daniels.