Andrew also told BBC television that he had "let the side down" by staying with U.S. financier Jeffrey Epstein after Epstein's conviction for paying a teenage girl for sex.
One of Epstein's accusers Virginia Giuffre has said she was forced to have sex with Andrew in London, New York and on a private Caribbean island between 1999 and 2002, when she says Epstein kept her as a "sex slave".
In an excerpt of an interview given by Andrew to the BBC's Newsnight programme, the prince was asked about the claim by Giuffre that she had sex with him in a house in London.
"I have no recollection of ever meeting this lady, none whatsoever," Andrew said.
Andrew, 59, is the second son of Queen Elizabeth.
He married Sarah Ferguson in 1986 and the couple divorced 10 years later although they remain close friends.