A spokeswoman for the borough said Cllr O'Hare would continue with her normal duties until the investigation was concluded in the next two weeks.
She said: "The complaint was that the resident believed the email to be racist and offensive and not something that should be sent from a councillor. Clearly the email is racist and not what we should be using council equipment for. The council is taking this very seriously."
She added: "If a councillor sends an email that is racist then that's a breach of the members code of conduct and the council's own equal opportunities policy."
The spokeswoman said the investigation was being conducted internally, by the borough's Labour party.
She said at this stage the matter had not been referred to the local government Standards Board.
She said: "We are satisfied that the picture did not originate from within the council. It appears to have been an isolated incident, and the only people who have that picture are those who need it for the investigation."