The Sun has published videos of the England international defender allegedly exposing himself in a Manchester bar.
The 32-year-old also faces accusations of having groped one of his female friends that was in attendance.
Cheshire police have opened an investigation into the incidents in question, with the BBC quoting them as saying that they are "aware of a video circulating on social media" and that "enquiries are in the early stages and no arrests have been made at this time".
Reports on Thursday suggested that City were prepared to take matters further by removing Walker from their ranks.
It is, however, understood that no such decision has been made.
Walker’s contract with the reigning Premier League champions is due to run until the summer of 2024, with no extension talks having taken place as yet.