Federal investigators uncovered the recording during raids on Cohen's home and offices in April.
The FBI investigation stemmed in part from a referral by the USA special counsel's office, which is looking into possible coordination during the election campaign between Trump's aides and Russian officials. The FBI investigation is separate from Mueller's probe.
In a tweet, Trump called such taping "totally unheard of & perhaps illegal". Speaking to the NYT, he said that the recording, less than two minutes long, would show that while the discussion of payments did occur, no payment was ever made and would prove that the current president did nothing wrong.
The recording of the President - if done by Cohen in NY - would be legal, as the Empire State has laws which only require one party on a phone call or conversation to consent to any recording.
Mr. Trump also said in his tweet Saturday that a lawyer recording a client is "totally unheard of & perhaps illegal".
Ms McDougal says she had a 10-month affair with Mr Trump in 2006, a year after he Wednesday Melania Trump.
Ms Daniels says she and Mr Trump had sex in a hotel room in Lake Tahoe, a resort area between California and Nevada, in 2006.
Trump's personal lawyer Rudy Giuliani told CNN that Trump had no idea he was being recorded during his conversation with Cohen, which took place in the then-candidate's office at Trump Tower in September 2016.
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AMI, American Media Inc., is the parent company of the National Enquirer. The deal effectively ensured McDougal's silence for the remainder of the presidential campaign. And Michael Cohen's lawyers apparently didn't leak this thing to the New York Times today. The woman has claimed she had an affair with Trump.
"Inconceivable that the government would break into a lawyer's office (early in the morning) - nearly unheard of", Trump wrote on Twitter Saturday morning.
"They didn't break in, they had a warrant issued by a judge".
"Obviously, there is an ongoing investigation and we are sensitive to that", Davis said.
Culhane said Trump has been able in the past to dismiss these kind of stories as "fake news".
He added: "When the recording is heard, it will not hurt Mr Cohen".
The timing of the conversation between the GOP nominee and his longtime "fixer" also provides more evidence that Cohen was trying to squash embarrassing stories about Trump before the election - a major focus of the investigation by federal prosecutors in Manhattan.
But he told ABC News in an interview broadcast this month that he now puts "family and country first" and will not let anyone paint him as "a villain of this story".