Donald Trump appeared familiar with a deal that a Playboy model made to sell the rights to her story of an alleged affair with him when Trump discussed the matter in September 2016 with his attorney Michael Cohen, according to a transcript of their conversation.

The transcript, which was provided by President Trump’s legal team, shows that the then-GOP presidential nominee does not register confusion or surprise when Cohen refers to a plan to purchase the rights to model Karen McDougal’s story from American Media Inc., the parent company of the National Enquirer.

AMI, whose chief executive is David Pecker, had signed a $150,000 deal with McDougal in August 2016 for her story, but never published it.

“Um, I need to open up a company for the transfer of all of that info regarding our friend David,” Cohen says, according to the transcript.

“So what do you got to pay for this . . . one-fifty . . .” Trump says.

“Yes,” Cohen replies. “Um, and it’s all the stuff.”

On Tuesday night, Trump attorney Rudy Giuliani disputed the idea that the recording shows that Trump knew about the McDougal deal.

“It doesn’t,” he said in an interview with The Washington Post. “That’s open to interpretation, and we can have a fight about that.”

“To me it sounds like Cohen is explaining something to [Trump] that he doesn’t understand,” Giuliani said, adding: “He doesn’t seem that familiar with anything. There is nothing to indicate he knew anything in advance.”

The president’s legal team released the transcript after one of Cohen’s attorneys, Lanny Davis, provided CNN with the audio of the tape – marking a dramatic escalation of rhetoric and turn against Trump by the Cohen camp.

Davis told The Post that Cohen, who is under federal investigation for possible bank fraud and election law violations, decided to release the tape “following extensive discussions” with his lawyers.

The Cohen team was pushed to act by statements Giuliani had made about Cohen’s role in the conversation, Davis said.

“It became necessary to rebut false statements,” he said. “We were not going to let Michael become a punching bag.”

When asked whether the Cohen team would release more recordings of Trump in the coming weeks, David said he would not comment, but acknowledged, “There are a lot of other tapes.”

Trump’s legal team does not believe that he was captured saying anything of substance on any other recording made by Cohen, according to people familiar with the material.

The Wall Street Journal first reported four days before the November 2016 election that McDougal had been paid by the National Enquirer. At the time, Trump campaign spokeswoman Hope Hicks called McDougal’s claims “totally untrue.”

“We have no knowledge of any of this,” she said.

The revelation of the existence of the recording last week showed for the first time that Trump knew of her claims and efforts to keep her quiet at least two months earlier.

The transcript and audio released Tuesday indicate that Trump’s knowledge of McDougal’s claims to AMI predated that early September 2016 conversation with Cohen.

In the audio released by CNN, Trump sounds matter-of-fact as Cohen walks through his plan, saying at one point, “Yeah, I was thinking about that.”

There is no sign that Cohen followed through with the plan to purchase the rights to McDougal’s story from AMI.

However, the timing of the conversation between the GOP nominee and his longtime “fixer” also provides more evidence that Cohen was trying to quash embarrassing stories about Trump before the election – a major focus of the investigation by federal prosecutors in Manhattan.

In rival cable news appearances Tuesday, Cohen and Trump’s attorneys clashed over who on the tape suggested the transaction be completed with cash.

On the audio released by Cohen, Trump says something muffled and then can be heard saying, “pay with cash.”

In the transcript released by the president’s legal team, Trump is quoted as saying, “Don’t pay with cash.”

Davis zeroed in on Trump’s use of the word “cash” to suggest the then-candidate was seeking to mask his activities – a claim that Giuliani fiercely disputed.

“It’s outrageous that someone would tape his client surreptitiously and it’s also foolhardy for them to yell and scream and make believe what’s on the tape,” he said on Fox News.

Alan Futerfas, an attorney for Trump Organization, said Tuesday that the context of the word “cash” in the conversation is a reference to paying one time with liquid currency – rather than financing a transaction with a loan. He said the reference is clear because Trump was raising the idea of financing as another option.

“The notion that they were discussing using a bag of cash or green currency is ridiculous,” Futerfas said.

The transcript and the audio also show that Cohen told Trump he had discussed buying the rights to McDougal’s story with Allen Weisselberg, the chief financial officer of the Trump Organization.

Futerfas disputed that Tuesday.

“The notion that Mr. Cohen would have spoken to Mr. Weisselberg about a proposition he had yet to even make to the president does not ring true,” he said. “Mr. Weisselberg is a bookkeeper who simply carries out directions from others about monetary payments and transfers. There would be no reason for Mr. Cohen to have any conversation with Mr. Weisselberg prior to him recommending and obtaining approval for the purchase he was suggesting. ”

The Post’s Philip Rucker contributed to this report.