It will be broadcast on BBC Two at 21:00 GMT on Saturday.
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.
In 2010, the prince was photographed walking with Epstein in New York’s Central Park – two years after Epstein’s first conviction for soliciting a minor for prostitution.
Footage published by the Mail on Sunday showed the prince in Epstein’s Manhattan mansion at about the same time.
Challenged on his decision to stay at the home of a convicted sex offender, the prince said: “It was a convenient place to stay.
“I mean I’ve gone through this in my mind so many times. At the end of the day, with a benefit of all the hindsight that one can have, it was definitely the wrong thing to do.
“But at the time I felt it was the honourable and right thing to do and I admit fully that my judgement was probably coloured by my tendency to be too honourable but that’s just the way it is.”
In 2015, Prince Andrew was named in court papers as part of a US civil case against Epstein.
One of Epstein’s accusers, Virginia Roberts – now Virginia Giuffre – said she was forced to have sex with the prince three times between 2001 – when she was 17 – and 2002, in London, New York and on a private Caribbean island owned by Epstein.
At the time she was under-age according to Florida state law.
The prince replied: “I have no recollection of ever meeting this lady, none whatsoever.”
When asked once more whether he remembered meeting Ms Giuffre, the prince said: “No.”