Mr Bowkett is a member of the Liberal Democrat group, but the role of city mayor is apolitical and ceremonial.
A spokesperson for the Liberal Democrat group said the video is “appalling” and they urged the council to launch a formal complaints and standards investigation.
“Gloucester deserves councillors whose priority is the city and the residents they serve.
“Not cheap stunts. Not mockery. Not behaviour that belongs in social media outrage clips,” they added.
When asked whether a formal complaint had been lodged against Mr Chambers, Gloucester City Council said it “cannot comment” on “individual elected members.”
A special meeting in to the authority’s financial position is scheduled to take place next week.