Answering questions about his links to convicted sex offender Jeffrey Epstein in a BBC interview, Prince Andrew said the alleged incidents "never happened".
Virginia Giuffre, one of Epstein's accusers, claimed she was forced to have sex with the prince three times.
The prince said he was at home with his children on one of the occasions.
Prince Andrew, who is the Queen's third child, has been facing questions for several months over his ties to Epstein, a 66-year-old American financier who took his own life while awaiting trial on sex-trafficking charges.
