The lawyer for porn star Stormy Daniels told CNBC on Friday that he knows secret recordings reportedly made by President Donald Trump's ex-attorney Michel Cohen about paying off a Playboy model are "the tip of the iceberg."

That lawyer, Michael Avenatti, had alleged the existence of Cohen's recordings in May during court proceedings involving materials seized from Cohen in FBI raids a month earlier.

"Back on May 30, in front of the federal courthouse, I demanded the release of the Trump tapes and disclosed their existence at that time, and I am once again demanding the release of all audio recordings made of Donald Trump," Avenatti said.

At that time, Avenatti suggested the recordings could potentially relate to his client Daniels, who is suing Trump and Cohen to void a contract she signed that prevents her from discussing an alleged affair with Trump from years earlier.

When asked whether he believes tapes made by Cohen related to Daniels exist, Avenatti indicated that he did.

"I think this is the tip of the iceberg," he said. Asked to clarify, Avenatti rephrased: ”I know this is the tip of the iceberg."

Avenatti was reacting to a new report from The New York Times on Friday that said Cohen recorded a conversation with Trump two months before the 2016 election in which Trump and Cohen discussed payments to ex-Playboy model Karen McDougal. She had alleged she had an affair with Trump.

A lawyer for Cohen, as well as Trump's outside counsel, Rudy Giuliani, did not immediately respond to CNBC's requests for comment on Avenatti's claims.

But a person familiar with the president’s legal strategy told CNBC that Trump's legal team is aware of other recordings, though not of any "substantive" tapes. The person added that Trump was "unaware" that he was being recorded during the reported discussion related to McDougal.

One of the reporters on the Times' story, Maggie Haberman, tweeted that the McDougal tape "appears to be the only one" between Trump and Cohen and adding in a follow-up that she's "not sure" how Avenatti "would know" that there are multiple recordings.

Haberman tweet

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Asked about Haberman's tweet, Avenatti said: "There is definitely more than one."

In an interview with MSNBC's Andrea Mitchell shortly after the Times published its story, Avenatti declined to detail his knowledge of what he called "multiple audio recordings."

He also called Cohen "one of the world's great evidence hoarders," which he characterized as being a "very, very bad thing for the president."

--CNBC's Eamon Javers contributed to this report.