The prince, 59, denies an allegation that he had sex with a 17-year-old girl procured for him by his former friend, who killed himself in a USA prison in August while awaiting trial on sex trafficking charges.
"It was a really scary time in my life", Ms Giuffre, previously known as Virginia Roberts, says in the teaser. And there's only one of us telling the truth'.
The Queen has been lightening her load in recent years and her husband, Prince Phillip, already withdrew from his royal duties two years ago, when he was 95 years old.
Ms Giuffre - then called Virginia Roberts - alleges she was forced to have sex with Prince Andrew three times between 2001 - when she was 17 - and 2002, in London, New York and Epstein's private island in the US Virgin Islands.
Andrew, 59, has repeatedly denied the allegations, most recently during a televised interview broadcast almost two weeks ago in which he lost public support by defending his friendship with Epstein and by not expressing sympathy for Epstein's many young female victims.
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The hour-long special, which airs on BBC One on Monday was recorded before the Duke's disastrous Newsnight interview, in which he denied the allegation.
British outlet the Daily Mail says Charles met with Prince Philip, Duke of Edinburgh, yesterday at Sandringham to discuss the move, which comes on the back of Charles' role in forcibly "retiring" Prince Andrew from public life.
In July 2015, Ms Giuffre went to the Metropolitan Police with her claims, but was ultimately turned away.
As the story dominated British news for a week, overshadowing campaigning for the December 12 election, there has been widespread criticism of Andrew and demands he should answer questions from USA authorities about Epstein.
The prince has since stepped down from royal duties because of the scandal.
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The Metropolitan Police's Murray said police reviewed its decision after Epstein's death and 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.
He added: "Officers assessed the available evidence, interviewed the complainant and obtained early investigative advice from the Crown Prosecution Service".
We therefore concluded that the MPS [Metropolitan police service] was not the appropriate authority to conduct enquiries in these circumstances.
"Following the legal advice, it was clear that any investigation into human trafficking would be largely focused on activities and relationships outside the UK" he said.
Epstein killed himself in prison while awaiting trial on charges of conspiracy and trafficking minors for sex.
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"Charles has had discussions with William", the source said.