Gloria Allred's comments come as the Duke has been targeted in a bus stunt organised by a US lawyer representing some of the victims of sex offender Epstein.
She told reporters: "I can assure Prince Andrew, it's not over until it's over, it's not over yet."
"You must do the right thing and stop shaming your family, the Queen, your children and your reputation."
She urged the Duke: "If you've done nothing wrong then just talk to the FBI, if you have done something wrong it's time to confess and tell us what that is."
Ms Allred said she wrote to the Duke of York last year, after he stepped down from royal duties, asking him to communicate immediately with the FBI and submit a sworn deposition.
Last month, the chief federal prosecutor in New York, Geoffrey Berman, said the FBI had received "zero co-operation" from Prince Andrew’s legal team despite their requests.
They want the Queen’s second son to assist them in their investigation into the sex offender Jeffrey Epstein.
Ms Allred continued: "We are not going to stop reminding you that you haven't spoken to the FBI until you finally do.
"The children and the adult survivors of sexual abuse deserve it."
Ms Allred alleges Prince Andrew was "part of the manipulation" of Epstein.
She said that "he has information which is definitely relevant".
She claims Epstein used the Prince’s name when trying to lure young girls to his private island.
Ms Allred added that the Duke's "silence is a statement and it’s not a positive statement for him".
Virginia Giuffre, who claims she was trafficked by Epstein, said in an interview with BBC Panorama that she was left "horrified and ashamed" after an alleged sexual encounter with Andrew in London in 2001.
Ms Roberts-Giuffre claimed she danced with Andrew in Tramp nightclub, adding he was "the most hideous dancer I've ever seen in my life" and "his sweat was... raining basically everywhere".
The prince, said in December, that he can "absolutely and categorically" say "it never happened".
During his car crash interview on BBC Newsnight he said he does not recall meeting Ms Giuffre and has strongly denied having sex with her – claiming a medical condition that meant he did not sweat.
Asked about a photo that appears to show him with his arm around Ms Giuffre’s waist, the prince said he didn't recall the photograph ever being taken and questioned if it was his hand in the picture.
Andrew has consistently denied accusations he had sex with any underage women.
His representatives have yet to comment on the latest development.