Late on Sunday night NZ time, Trump took to Twitter, saying "that the government would break into a lawyer's office (early in the morning) - nearly unheard of".
On Friday, three people with knowledge of the conversation told The Post that Cohen had secretly taped a conversation with Trump in September 2016 about whether to purchase the rights to the story of one of those women, Playboy centerfold Karen McDougal.
Trump's attorney Rudy Giuliani confirmed to The Times that Trump discussed payments to McDougal with Cohen, but he said that ultimately no payment was made.
The tapes were reportedly discovered during an Federal Bureau of Investigation raid on Mr Cohen's property earlier this year in NY. The practice, known as "catch and kill" kept the story from becoming public for the remainder of the presidential campaign.
That payout seems to be a significant part of a federal investigation into Cohen that's been going on for months, as the hush money may be in violation of campaign finance laws. Giuliani says what's important is Trump said that if it were made it should be properly documented.
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"I think we need to bring this in-house", Cohen tells Trump, according to one person with knowledge of the recording. This gives the FBI access to the recording as part of the USA attorney for the Southern District of NY investigation into Cohen.
The US president was responding to reports he had been recorded by Michael Cohen discussing payments to a Playboy model, who said she had an affair with Mr Trump. "Any attempt at spin cannot change what is on the tape". Cohen's attorney, Lanny Davis, told CNN in a statement the recording will not hurt Cohen.
But he said no money was actually ever transferred, and the recording itself amounted to "powerful exculpatory evidence".
In her lawsuit against AMI, McDougal said she was happy when AMI bought her story then did not publish it, because she was not anxious for publicity. The Playboy model in question is Karen McDougal, who has claimed she had an affair with Trump. The Trump administration had previously denied knowing anything about such a payment.
When informed about the tape, the president said "I can't believe Michael would do this with me", according to a source familiar with the tapes.
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.
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 USA campaign finance laws.