(CNN) President Donald Trump's lawyers have waived attorney-client privilege on his behalf regarding a secretly recorded conversation he had in September 2016 with his former longtime lawyer Michael Cohen in which they discussed payments to an ex-Playboy model who says she had an affair with the President, according to sources familiar with the matter.

The move comes as an attorney for Cohen openly questioned Trump lawyer Rudy Giuliani's claim that the tape showed no wrongdoing by the President, furthering the growing divide between Trump and Cohen, who had once said he'd take a bullet for the President.

Trump's lawyers asking on behalf of the President to remove the privilege designation from the recording means that the government now has access to it as part of the US attorney for the Southern District of New York's probe into Cohen. It effectively gives prosecutors the ability to use the recording if they find it relevant to their criminal investigation of Cohen.

The prosecutors working on the case had not reviewed the recording because it, along with millions of other documents and files seized in FBI 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 special master had designated the recording as privileged, according to two sources familiar with the process, but Trump's lawyers subsequently waived their right to maintain that designation.

After The New York Times first revealed the existence of the tape -- and Giuliani claimed the tape was exculpatory evidence that did no harm to Trump — his legal team decided to remove any protection relating to the attorney-client relationship on this specific matter.

The President's attorneys clearly believe that the tape will not show any wrongdoing by their client. The transaction never happened, Giuliani told CNN.

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." Later Lanny Davis, Cohen's newest attorney, put out a statement claiming that "when the recording is heard, it will not hurt Michael Cohen. Any attempt at spin cannot change what is on the tape."

Team Cohen remains surprised that Trump's lawyers removed the protective claim of privilege, given their view that the tape is harmful to the President. They claim that Giuliani is mischaracterizing the tape in key ways that contradict the facts of the conversation as they see them. But they declined to offer a description of their exact view of the conversation beyond saying that Giuliani's more benign descriptions of the tape are contradictory to what actually occurred.

What's more, they see Giuliani's public comments as a way to change the subject from the President's near-universally panned performance in Helsinki, Finland, at the beginning of last week.

Giuliani put a positive spin on the content of the in-person conversation on Friday, describing it as Trump discussing potential payments to former Playboy model Karen McDougal, who alleges Trump an affair with her. Trump denies the allegation.

"Cohen is talking about buying the rights from AMI (American Media Inc.)," Giuliani told CNN on Friday. "They're talking about a corporation doing it, one of their corporations doing it. The President says 'make sure it's done correctly, and make sure it's done by check.'"

Giuliani's point is that Trump did nothing wrong -- that before the conversation he knew nothing about any payments, noting that no payment was made and that he suggested paying by check so there would be a record of it.

The President's lawyer also sought to position the President as wronged by the recording. Trump was stunned to hear about the tape, Giuliani said, and, according to a source familiar with the matter, responded when informed of its existence that "I can't believe Michael would do this to me." Giuliani said the recording is short—less than two minutes, according to another source -- and was simply a professional exchange between a client and his lawyer.

That was echoed Saturday by the President, who tweeted that it was "inconceivable that a lawyer would tape a client - totally unheard of & perhaps illegal. The good news is that your favorite President did nothing wrong!"

In response, Cohen attorney Davis tweeted, "The strategy of @realdonaldtrump @potus @RudyGiuliani is flawed; just as is #Trump's false #Twitter statement made against @michaelcohen212 this morning. Rudy claims the tape is "exculpatory". Why so angry?"

Asked about the Davis tweet, Giuliani told CNN on Saturday: "How would you feel if your lawyer was recording you without telling you? The tweet is straightforward and correct."

One of the sources familiar with the matter pointed out that New York is a "one-party" consent state, meaning that as long as one participant in the conversation gives permission to record, it is legal.

"Taping a conversation is the functional equivalent of retaining notes," the source said.

The source added that the recording and any lawyer's notes are privileged and non-public and called it ironic that Trump would complain about the recording and yet also choose to release it.

"It is the client who owns the privilege and not the lawyer," the source said. "In this specific instance, it was Trump who waived the privilege after Special Master Jones ruled the tape was privileged. Team Trump actually contacted the judge and waived the privilege, thus, permitting Rudy (Giuliani) the ability to release his version of the tape's content. 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. Very foolish strategy by team Trump."

When the existence of the apparent deal with AMI was first reported by The Wall Street Journal just days before the election, the campaign said that the President was unaware of anything relating to a McDougal payoff, and Hope Hicks—then a spokeswoman—denied that Trump even had an affair with McDougal. But the recording, made two months earlier, suggests otherwise.

In addition, Giuliani has said there is only one real conversation with the President and the rest are non-substantive, short, more mundane conversations about ministerial matters, like setting up discussions.

But one source close to Cohen, without specifying any conversation or naming Trump himself, says there are "definitely all kinds of tapes out there." Two other sources said that beyond the tape with the President there were other recordings of "powerful" people speaking with Cohen that the FBI seized.