In her book, "Full Disclosure," Daniels also provides a graphic description of Trump's genitalia, including a reference to a toadstool-shaped character on the Mario Kart videogame, the British newspaper said.
The Guardian said it had obtained a copy of the book ahead of its publication on October 2, a month ahead of the key US midterm elections.
Trump has denied the affair but his lawyer Michael Cohen has pleaded guilty to campaign finance violations linked to $130,000 in "hush money" he paid Daniels shortly before the November 2016 presidential election.
Daniels claims to have had a sexual liason with Trump at a celebrity golf tournament in Lake Tahoe in 2006, around the time that the reality television star's wife, Melania, had given birth to their son Barron.
In the book, she describes seeing Trump for the first time at the golf tournament.
"He had a red cap, a Trump crest as a placeholder for the MAGA slogan not one of us could see coming," Daniels writes.
She said they had sex in Trump's penthouse after one of his bodyguards invited her to have dinner with the real estate tycoon.
"It may have been the least impressive sex I'd ever had, but clearly, he didn't share that opinion," she writes.
In the book, Daniels also recounts her mounting disbelief in 2016 as Trump closes in on the Republican presidential nomination, The Guardian said.
"It will never happen, I would say," Daniel writes. "He doesn't even want to be president."
Daniels said she remained in touch with Trump over the next year in the hope he would get her on his reality television show, The Apprentice.
She said Trump even suggested that he would allow her to cheat to remain on the show for more episodes.
"He was going to have me cheat, and it was 100 percent his idea," the Guardian quoted her as saying in the book.
Michael Avenatti, Daniels' lawyer, tweeted following the Guardian report that the "most important thing about (the book) is not the description of her sex with Mr. Trump."
"It is instead her description of her life and role as a modern woman unafraid to speak truth to power," Avenatti said.