In a special hour-long Panorama, to be aired on BBC One on 2 December but recorded before the Newsnight interview, Ms Giuffre says: "It was a really scary time in my life".
In the interview, the duke denied Ms Giuffre's claims and said an alleged encounter with her in 2001 did not happen as he spent the day with his daughter, Princess Beatrice, taking her to Pizza Express in Woking for a party.
Virginia Roberts, who claims she was trafficked by convicted pedophile Jeffrey Epstein and forced to have sex with Prince Andrew on three occasions almost 20 years ago, has responded to the prince's denial of her allegations against him by saying: "He knows what happened, I know what happened, and there's only one of us telling the truth".
Ms Giuffre alleged the duke sweated heavily as they danced at London nightclub Tramp.
Murray also said that London police have not received a formal request for assistance from other law enforcement agencies investigating the case.
Reports emerged today that Prince Andrew allegedly smuggled a "masseuse" into Buckingham Palace after being introduced to the woman by Epstein's alleged "madam" Ghislaine Maxwell. I know what happened. "And there's only one of us telling the truth", she added.
Buckingham Palace has described the allegations as "false and without any foundation", stating: "Any suggestion of impropriety with under-age minors" by the duke was "categorically untrue".
Earlier on Thursday, the Metropolitan Police defended its decision not to investigate a claim of trafficking by Ms Giuffre. Breitbart News reported that the son of Queen Elizabeth II flew on Epstein's private plane on at least two occasions with Virginia Roberts, then 17-years-old.
Prince Andrew appears on BBC Newsnight with Emily Maitlis in regard to the Jeffrey Epstein scandal. The allegation was made against a U.S. national, Jeffrey Epstein, and a British woman.
He said the decision was reviewed after Epstein's death and police decided not to change policy. "It related to events outside of the United Kingdom and an allegation of trafficking to central London in March 2001", United Kingdom police said in a statement.
London's Metropolitan Police Service (MPS) said it had received an allegation in 2015 of non-recent trafficking against Epstein and a British woman whom it did not name.
"We therefore concluded that the MPS was not the appropriate authority to conduct enquiries in these circumstances and, in November 2016, a decision was made that this matter would not proceed to a full criminal investigation", Alex Murray said.
Epstein died August 10 after being found unconscious in a Manhattan jail cell, where he was awaiting trial on federal sex trafficking charges.