Ms. Clifford said during the interview that while she had seen Mr. Trump more than once, she had had sex with him a single time, unprotected. That happened shortly after they met at a celebrity golf tournament in Lake Tahoe in 2006. (Ms. McDougal has also said that she, too, was intimate with Mr. Trump during that event.) Mr. Trump was 60 at the time; Ms. Clifford was 27.

Ms. Clifford said that Mr. Trump had invited her to his hotel suite for dinner, and that their banter began with him showing her a magazine cover featuring his photograph. “I was like, ‘Someone should take that magazine and spank you with it,”’ she said. “So he turned around and pulled his pants down a little — you know, had underwear on and stuff, and I just gave him a couple swats.”

It was done in a joking manner, she said, and the flirtation — which included Mr. Trump comparing Ms. Clifford to his daughter — led to intercourse, though, she said, she had not been particularly attracted to Mr. Trump and had not wanted to have sex with him. (She said she nonetheless went along of her own accord.)

“He said that it was great,” she said, and told her he had had “a great evening, and it was nothing like he expected, that I really surprised him, that a lot of people must underestimate me — that he hoped that I would be willing to see him again, and that we would discuss the things we had talked about earlier in the evening.”

Mr. Trump, she said, had raised the possibility that he get her onto his reality show, “Celebrity Apprentice,” but it would not come to be.

She said that he had invited her to his Beverly Hills Hotel bungalow in 2007 to fill her in on that promise. The entertainment for the night, she said, was a “Shark Week” documentary. Mr. Trump, she said, wanted to have sex but this time she did not go along with it, and when he did not have an answer for her about being on the show, she left. She said he told her over the phone a month later that he would not be able to get her onto the program and that was effectively the end of it.

But when her story threatened to surface again in 2016, Ms. Clifford said, her lawyer, Keith Davidson, called her. “I think I have the best deal for you,” she said Mr. Davidson told her, presenting Mr. Cohen’s offer. (Mr. Davidson had also represented Ms. McDougal.)