The royal has come increasingly under fire over his relationship with Epstein, who died in his prison cell in August as he awaited trial on sex-trafficking charges.
Now he has said he kicks himself over the decision to stay with the sex offender, saying that it was not “becoming” of a member of the royal family.
Speaking to BBC News, he said: “The problem was the fact that once he had been convicted I stayed with him.
“That’s the bit that I kick myself for on a daily basis, because it was not something that was becoming of the royal family.
“We try to uphold the highest standards and practices and I let the side down.”
The 59-year-old royal was accused of taking part in an orgy with nine girls by Virginia Roberts Giuffre, who was pictured with his arm around her waist when she was 17 years old. She has claimed that the pair had sex on three occasions.
Commenting in an interview with the BBC – part of a Newsnight special due to air on Saturday – Prince Andrew said: “I have no recollection of ever meeting this lady, none whatsoever.”
Presenter Emily Maitlis went on to repeat the claims by Mrs Giuffre, then known as Virginia Roberts, that “she met you in 2001, she says she dined with you, danced with you at Tramp Nightclub in London. She went on to have sex with you in a house in Belgravia”.
Ms Maitlis added: “You don’t remember meeting her?”, to which Andrew replied: “No.”
Buckingham Palace has branded the allegations “false and without any foundation” and said “any suggestion of impropriety with underage minors” by the duke was “categorically untrue”.
‘Prince Andrew & the Epstein Scandal: The Newsnight Interview’ airs Saturday on BBC2 at 9pm