The income went to the former prince rather than his landlord the Crown Estate.
Mountbatten-Windsor has been summonsed to give evidence in a businessman's High Court fraud trial.
The force said it had received information and stressed its inquiries were at "an early stage".
Scottish police are investigating the time convicted sex offender Jeffrey Epstein spent with Andrew Mountbatten-Windsor at Balmoral, a private royal estate in Scotland which is a favored retreat for the monarchy.
World News
The High Court trial involves allegations of fraud made by a wealthy Turkish woman against her former adviser.