Twelve audio recordings seized from US President Donald Trump's onetime personal attorney, Michael Cohen, were turned over to federal prosecutors on Friday, according to a filing in federal court in Manhattan on Monday.
Mr Giuliani said Mr Cohen had recorded the McDougal conversation at Trump Tower in NY using a hidden device.
Now, the recording is available for the world to listen to.
Cohen's lawyer went on CNN to discuss the tapes, citing Trump suggesting cash to make a payment in the audio as something only "drug dealers and mobsters" use.
In the at-times muddled recording, Mr Cohen says he wants to set up a company to acquire the rights off American Media, the company that owns the Enquirer.
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Cohen can be heard in the tape telling Trump that he needs "to open up a company for the transfer of all of that info regarding our friend David", apparently referring to American Media Inc. head David Pecker.
McDougal later sued, claiming that AMI paid for the story with the intention of burying it to protect Trump.
In the tape, Trump and Cohen talk about buying the rights to McDougal's story, which she had sold a month earlier to the National Enquirer for US$150,000. When it comes time for the financing, which will be. In Azerbaijan, Mr Trump did business with a figure linked to money laundering and the Iranian Revolutionary Guard.
Trump responded to the release of the tape by trying to undermine Cohen's professionalism.
Trump's new attorney Rudy Giuliani claims Trump said "don't pay with cash", sticking with his previous claims that Trump wanted everything on paper. Mr Cohen: "No, no".
CNN reported that Trump's attorneys had waived his attorney-client privilege on the tape.
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Caputo admitted to being "shocked" upon discovering Cohen had secretly recorded conversations with Trump but did not anticipate any legal landmines.
Although it is unclear if the content of the tape creates legal hurdles for the president regarding campaign finance violations, Trumpers believe the pair's conflict has been orchestrated by Special Counsel Robert Mueller's strategy for the Russian Federation probe.
"Suggesting otherwise is ridiculous and is inconsistent with the rest of the conversation, during which it was discussed doing it through a corporation", he said.
Cuomo aired a recording Tuesday night that Cohen secretly taped of a September 2016 conversation he had with Trump in which they discussed buying another woman's story of an alleged affair.
Mr Cohen says: "We'll have to pay".
Stormy Daniels' attorney Michael Avenatti urged Cohen again on Wednesday to immediately release "all" the tapes, other evidence in his possession and "come clean" about his conduct and that of the president. The conversation about the potential payment appears to have been interrupted in the middle of Trump saying what sounds like "check", and abruptly cuts to a different person for a few seconds.
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