"Inconceivable that the government would break into a lawyer's office (early in the morning) - nearly unheard of", Trump tweeted.
In his first public comments since reports were published on the recording in The New York Times and Wall Street Journal on Friday, Trump said it was "inconceivable that a lawyer would tape a client - totally unheard of & perhaps illegal".
In the brief recording made in September 2016, Cohen can be heard telling Trump that AMI had recently purchased the rights to McDougal's account of a 10-month affair that allegedly took place soon after he married Melania.
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.
The taped conversation, now in the FBI's possession, could also further entangle the president in a criminal investigation that for months has targeted Cohen.
"Nothing in that conversation suggests that he had any knowledge of it in advance", Giuliani told the newspaper of American Media Inc.'s payment.
It 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 United States campaign finance laws.
NY law allows one party to a conversation to tape it without the other knowing.
The president has said that Mr Trump is no longer his personal lawyer.
Trump is largely silent in the conversation, the people said, neither expressing surprise nor indicating whether he knew previously about the AMI deal. There are conflicting reports about what is on the tape, but Giuliani has confirmed its existence and the fact that it is a discussion of a payment.
Cohen's recording adds to questions about whether Trump tried to quash damaging stories before the election. "Any attempt at spin cannot change what is on the tape".
Cohen, who became Trump's personal lawyer in 2007, is infamous for paying $130,000 to Daniels - real name, Stephanie Clifford - shortly before the 2016 election to keep quiet about her alleged affair with Trump.
At this point, Cohen has not has any conversations with prosecutors in the Southern District of NY, one source says. It was not immediately clear where Trump and Cohen were located at the time of the call.
McDougal says she began a almost yearlong affair with Trump in 2006, shortly after Trump's wife, Melania, gave birth to their son Barron. But the agreement meant that McDougal signed a non-disclosure agreement that prevented her from revealing the affair. But the tabloid did not publish the story, keeping it out of public view.