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Adult film star Stormy Daniels — who has said in a lawsuit that she had an "intimate" relationship with Donald Trump a decade before he became president — said in an interview airing Sunday night that she was threatened in 2011 inside a Las Vegas parking garage with her infant daughter present and was told not to go public with her story.

In a wide-ranging interview with CBS' "60 Minutes" — its highest-rated broadcast in 10 years, since an interview with Barack and Michelle Obama in 2008, the network said Monday — Daniels, whose real name is Stephanie Clifford, said the threat came weeks after she'd agreed to tell her story to a sister publication of InTouch magazine — an interview that didn't run at the time.

"I was in a parking lot, going to a fitness class with my infant daughter. Taking, you know, the seats facing backwards in the backseat, diaper bag, you know, getting all the stuff out. And a guy walked up on me and said to me, 'Leave Trump alone. Forget the story,'" Daniels said, according to a transcript of the interview.

"And then he leaned around and looked at my daughter and said, 'That's a beautiful little girl. It'd be a shame if something happened to her mom.' And then he was gone," she added.

Daniels said that she interpreted the event as direct threat and that she was "rattled."

"I remember going into the workout class. And my hands are shaking so much, I was afraid I was gonna drop her," Daniels said.

Daniels said that she didn't go to the police because she was scared and that she never again saw the man who threatened her, but she said she'd "100 percent" be able to recognize him if she ever saw him again.

"Even now, all these years later. If he walked in this door right now, I would instantly know," she said.

In a lawsuit filed earlier this month, Daniels said she had an "intimate" relationship with Trump in 2006 and 2007 and struck a deal a decade later to keep quiet about it. Daniels signed the deal, a secrecy agreement facilitated by Trump's personal attorney, Michael Cohen, in exchange for $130,000 before the 2016 election.

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She has contended, however, that the pact is void because Trump never signed the agreement.

NBC asked the White House and Cohen for comment on the "60 Minutes" report. Neither the White House nor Cohen responded to questions from "60 Minutes."

First Lady Melania Trump, when asked by NBC News whether she watched Daniels' appearance on "60 Minutes" or whether she had any comment about Daniels' allegations, said through a spokeswoman on Monday that she was focused on her son and "future projects."

"She's focused on being a mom and is quite enjoying spring break at Mar-a-Lago while working on future projects," Stephanie Grisham, a spokeswoman for Melania Trump said, making no reference to Trump.

Meanwhile, in a letter Sunday night to Michael Avenatti, an attorney for Daniels, an attorney for Cohen denied that Cohen "was responsible for an alleged thug" who may have made an "alleged threat" to Daniels.

The letter, which the lawyer, Brent Blakely, provided to NBC News, demands that Avenatti retract the accusation, apologize and "make clear that you have no facts or evidence whatsoever to support your allegations."

And Charles Harder, an attorney for Trump, asked "60 Minutes" to show on camera and read on air one of the statements Daniels signed in January, in which she denied reports that she had had an affair with Trump.

"My involvement with Donald Trump was limited to a few public appearances and nothing more," the statement read.

Daniels' account of the threat is the most harrowing yet in the saga over her efforts to go public about her alleged affair with Trump.

Nine days ago, Avenatti repeatedly said in an interview with MSNBC's "Morning Joe" that his client had been threatened, but he wouldn't say by whom. He also declined to give details on the threats.

The White House has denied that Trump had an affair with Daniels. White House Press Secretary Sarah Sanders said this month that, as far as she knows, the president was not aware that Cohen paid Daniels the $130,000.

Cohen has previously said that he used his own money to facilitate the payment and that "neither the Trump Organization nor the Trump campaign was a party to the transaction with Ms. Clifford, and neither reimbursed me for the payment, either directly or indirectly."

Daniels has offered to give back the money so she can speak freely about Trump and release any text messages, photos and videos she might have.

Daniels, however, wouldn't comment whether she had any video images, pictures or text messages to release.

"My attorney has recommended that I don't discuss those things," she said.

But she did share some of the more intimate details of her sexual encounter with Trump — an experience that happened only once, she said — including that he did not wear a condom.

"Did you ask him to?" interviewer Anderson Cooper asked.

"No. I honestly didn't say anything," Daniels said.