The Campaign group Republic has lodged a formal complaint with UK Trade and Investment (UKTI) calling for an investigation into a number of allegations relating to Prince Andrew.
The complaint raises Republic's concerns as to whether the Duke of York is damaging Britain's reputation, particularly in the light of recent reports of his friendship with the convicted sex offender Jeffrey Epstein and of his links to the Gaddafi regime in Libya.
The letter calls on the UKTI to follow the example of the LSE and investigate Andrew's relationship with Libya, as well as other serious accusations which have been made against him.
Republic spokesperson Graham Smith said: "It is clear that the continued speculation about the Duke's friendships, business interests and professional conduct risks bringing UKTI, the Department for Business and the Foreign Office into disrepute.
"We make no claims as to the veracity of the allegations against the Duke. We simply ask that they are treated in a manner that reflects their seriousness, as they would be if made against any other public servant."
A Downing Street source described the recent flurry of stories surrounding the Duke of York as the media trying to create "guilt by association" but added that one more serious story could change the situation overnight.
The Business Secretary Vince Cable said the Duke of York would have to judge his own position, although there would be "conversations" about his future role.