Ms Giuffre wrote the unpublished memoir about her life as a teenage victim inside the convicted paedophile Epstein's alleged sex-trafficking ring.
The royal categorically denies he had any form of sexual contact or relationship with Ms Giuffre.
Ms Giuffre, 36, claimed in her memoir that the incident took place in Easter 2001 when Maxwell called her to the first-floor study of Epstein's home in Manhattan, New York.
"Ghislane led me to the decadent sofa he was lounging on and twirled me around to give him a good look over before sitting me down on his lap," Ms Giuffre wrote.
"Like a show pony I knew exactly how they wanted me to be paraded around, with the last of my dignity long lost, I had nothing more to lose.
"It was easy to give them the reaction they desired, all I had too do was pretend to be entertained by their lewd gestures, and when Andrew cupped my breast with a doll made in his image, I only giggled away.
"Ghislane wanted to take a picture of the bizarre scene and even got Johanna (Sjoberg), another one of Jeffrey's so-called personal assistants, to come sit on his other knee for the snapshot, giving the impression girls couldn't stay away from Randy Andy."
Johanna Sjoberg, 40, has also made allegations of sexual misconduct against Epstein and Prince Andrew.
Ms Sjoberg previously described the encounter with the Duke of York and the puppet when testifying in a 2016 deposition in Ms Giuffre's civil defamation case against Maxwell.
"I sat on Andrew’s lap... they took the puppet’s hands and put it on Virginia’s breast, and so Andrew put his on mine," said Ms Sjoberg on oath.
Ms Giuffre goes on to describe how she was ordered to take Prince Andrew to the "dungeon" for an erotic massage in the memoir.
She wrote that she wanted to "run and hide" but her "servitude for Jeffrey" kept her in the room.
"I turned on the steam room and began to undress myself. Andrew followed my lead and disrobed, throwing his attire to the floor.
"Getting out of his stronghold I turned my attention to the steam filling up and tried to turn his attention to the marble bench where I suggested he take a seat while I rubbed his feet.
"By the look in his eyes I knew he could care less about a foot massage, his real desires were being shown by the arousal his body was forming.
"He indicated where he wanted me to touch him and how. I fulfilled his seedy requests, but only just."
Ms Giuffre added that "there was no pleasure" for her and the incident caused her to sink further into a depression.
A representative for Prince Andrew declined to comment on the claims.
US authorities have formally requested that the Duke of York answer questions as a witness in a criminal probe into sex offender Epstein.
But the Sun reports a decision on the request had yet to be made by British officials.
Maxwell is now in jail in the US on charges of trafficking minors for her former partner and the British socialite between 1994 and 1997. She has pleaded not guilty and is scheduled to go on trial in July 2021.
Prince Andrew has said he first met Epstein in 1999.
The disgraced financier was found hanged at age 66 last August in a Manhattan jail, while awaiting trial on sex trafficking charges for abusing women and girls in Manhattan and Florida from 2002 to 2005.
He had pleaded not guilty.