The policy of flying Union flags on royal birthdays is being re-examined after councils were ordered to do so on the Duke of York’s birthday.
Downing Street said the matter was being looked into after critics branded the reminder "crass and offensive".
An email, seen by the Sun, has been sent to councils by Matt Stevenson - private secretary to Ministry of Housing, Communities and Local Government permanent secretary Dame Melanie Dawes - reminding them to fly the flag for Andrew’s 60th birthday on February 19.
But critics said the order should be scrapped over the allegations Andrew slept with Virginia Giuffre, who claims she was trafficked by convicted US sex offender Jeffrey Epstein.
Prime Minister Boris Johnson’s official spokesman said: "This was an administrative email about long-standing policy."
Mr Johnson's spokesman added: "I understand DCMS (Department of Culture, Media and Sport) and the Royal Household are considering how the policy applies in changing circumstances such as when members of the royal family step back from their duties."
Liverpool Mayor Joe Anderson said Liverpool City Council will not be obeying any such requests from the Government because it would not be "appropriate".
Mr Anderson told the Liverpool Echo: "No, we won’t be doing that, I don’t think that would be appropriate.
"This isn’t to do with being anti-royal, we have flown the flag for the Queen before.
"But Prince Andrew isn’t a major royal, he’s not a significant member of the royal family."
Mr Anderson added: "When you look at his behaviour, it wouldn’t be appropriate for us to mark his birthday."
Andrew stepped down from royal public duties in November after the fallout from his disastrous Newsnight appearance.
He was accused of showing a lack of empathy for Epstein’s victims and of failing to show regret over his friendship with the disgraced financier.
Ms Giuffre, previously known as Virginia Roberts, says she and the duke slept together on three separate occasions, including when she was 17 – still a minor under US law.
Andrew, who strenuously denies the allegations, is facing calls to talk to the FBI and US prosecutors.
Certain royal birthdays, including Andrew’s, are also marked by the ringing of bells at Westminster Abbey in central London.
The re-examination of the flag-flying policy will not affect the Duke and Duchess of Sussex, who are also quitting royal duties amid the Megxit crisis.
Harry and Meghan's birthdays are not marked with flags on UK Government buildings, because the duke is not the child of a monarch, and the pair are not a future king and queen.
Republic, a campaign to replace the monarchy with an elected head of state, said councils had better things to worry about than "daft royal protocols".
Republic chief executive Graham Smith said: "This is crass and offensive, and even at the best of times it’s nonsense to be flying flags for Andrew’s or anyone else’s birthday.
"Councils are struggling to provide core services, they have better things to worry about than daft royal protocols.
"And to think we should be celebrating a man who has serious accusations of sexual assault hanging over his head is particularly appalling.
"It’s time we ended the whole nonsense of flying flags for royal birthdays."
The Local Government Association said councils will decide locally when to fly flags.
A spokesman for the LGA said: "Councils have their own flag-raising guidelines or protocols and will decide locally how to respond to any requests to fly particular flags on specific occasions."