The Ivorian international on Sunday morning posted several screenshots of the racist abuse sent to his social media accounts.
Police have ave encouraged Zaha to report the incident to his local police force but say they are yet to receive a formal complaint.
Zaha meanwhile has received support from his manager Roy Hodgson who said that both clubs are working on identifying the individual responsible.
"It is very saddening on the day of a game that a player wakes up to this cowardly and despicable abuse," said Hodgson.
"I think it is right that Wilf made people aware of it; I don't think it is something he should keep quiet about.
"I think it is very good that our club, Aston Villa and the Premier League are doing everything they can to find out who this despicable individual is and one can only hope that they will get identified and they will get called to account and they will pay for these actions.
"There is literally no excuse; there is no excuse at all."