Police are examining allegations that Andrew, 65, sought help from a police officer to discredit a woman who accused him of sexual abuse when she was a teenager.
Andrew, the younger brother of King Charles III, said on Friday last week he would no longer use his Duke of York title among others, following years of criticism about his behaviour and connections to the late United States sex offender Jeffrey Epstein.
Andrew, who will remain a prince, has consistently denied any wrongdoing.
A court ruling last year showed that the British government believed one of Andrew's close business associates was a Chinese spy. Andrew at the time said he had stopped all contact with the businessman.
"We agree and support the decision that the royal family and Prince Andrew have taken," Bridget Phillipson, Education Minister and a senior MP, told Sky News on Monday.
"We believe that's the right course of action."
When asked about calls for the removal of Andrew's prince title, Phillipson said it was not a matter for the government but one for the royal family to decide.
In Nobody's Girl, obtained by the BBC ahead of publication, Giuffre alleges she feared she might "die a sex slave" under Jeffrey Epstein's control and describes three alleged sexual encounters with Andrew, including one involving Epstein and several other young women.
The memoir also alleges Andrew correctly guessed Giuffre's age - 17 - when they first met, and includes details of encounters in London, New York, and on Epstein's private island.
On Sunday, London's police force said it was "actively looking" into the allegations in a Mail on Sunday report that Andrew had asked one of his personal protection officers in 2011 to dig up information about Giuffre.
A Buckingham Palace source said those allegations should be "examined in an appropriate way".
Giuffre died in Australia in April this year. She was 41.
