She says the $150,000 (£115,000) agreement gave the tabloid exclusive story rights and banned her from talking publicly about the alleged affair.
Trump had apparently been unaware of the recordings until he was told about them earlier this week.
President Trump on Saturday described former lawyer Michael Cohen's taping of a private conversation between them as "totally unheard of & perhaps illegal", but he reassured supporters that he has done nothing wrong. He said the conversation does not pose any legal jeopardy for the president.
But Giuliani's claims that the tape does Trump no damage was disputed by a source close to Cohen who said that Giuliani is "trying to say what is bad is good".
"In the big scheme of things, it's powerful exculpatory evidence", he added. Trump continues to deny that he has ever paid hush money to keep his affair quiet.
The Justice Department is investigating Cohen's involvement in paying women in order to hide stories about Trump before the 2016 Presidential election.
Cohen is under investigation for his business dealings and his handling of negative stories about the POTUS, including, allegedly paying porn star Stormy Daniels $130,000 in hush money to hide an affair. The discussed payment to AMI ultimately never happened, according to Giuliani.
It is ironic that Trump would complain about a privileged tape that would not have been released and then chooses to make it public himself.
"Nothing in that conversation suggests that he had any knowledge of it in advance", Giuliani told ABC News.
What has Trump previously said?In other words, they wanted to buy McDougal's silence. The two men discuss whether to use a check, rather than cash, which would create a record, according to the people.
"So not only is suggesting that Cohen committed a crime in recording him, but to accuse his former lawyer, who simply said may have the goods on him of committing the crime is a awful, bad idea", Specter continued.More news: 'Wild Boars' to miss out on live World Cup final
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The recording cuts off with Trump mid-sentence, one person said, and picks up in the middle of a conversation Cohen is having with another person. However, the episode raised the possibility that Mr Cohen secretly recorded other conversations with Mr Trump and senior campaign figures.
It's not yet clear exactly what the nature of the discussion was.
But his attorney, Lanny Davis, dismissed any notion that Cohen could be put in legal jeopardy due to the recording. "This is a very surprising tweet from the president this morning where he suggests his former lawyer, Michael Cohen, committed a crime". "Any attempt at spin cannot change what is on the tape".
He also called raids of Cohen's home, office and hotel room earlier this year "inconceivable" and "almost unheard of". "It is the client who owns the privilege and not the lawyer". That conversation took place weeks before Trump's November 2016 election.
Michael Cohen and Donald Trump.
"We have no knowledge of any of this", she said.
She claims that she started an affair with Trump in 2006, just after his wife, Melania, had their son, Barron. She turned down the offer and began a relationship that included, she said, interactions between the two "many dozens of times".
But the outlet never published the piece. But the agreement meant that McDougal signed a non-disclosure agreement that prevented her from revealing the affair.
She filed a lawsuit against AMI this year seeking to regain the rights to her story and settled with the company in April. But the Enquirer did not publish her kiss-and-tell, and she says she was tricked.
At this point, Cohen has not has any conversations with prosecutors in the Southern District of NY, one source says. The White House had also declined comment.