A video circulating on social media shows the midfielder preparing to take a corner at Stamford Bridge on Saturday, when one Chelsea fan began hurling abuse at the Scottish international.
Chelsea and Villa have both released statements on the incident as an investigation has been launched.
"We are aware of a video circulating on social media showing an individual using abusive language of a sectarian nature in the crowd at Stamford Bridge on Saturday," Chelsea said.
"Chelsea Football Club finds all forms of discriminatory behaviour totally unacceptable and we condemn the language used.
"The club is working to identify the person involved and will take the strongest possible action against them. Any supporter who was present in that area of the ground and has information that can assist the club’s investigation should contact the club via our discrimination reporting line 0207 3863355."
Villa added: "Aston Villa are grateful to our friends at Chelsea FC for their swift and decisive action in both condemning the bigotry aimed at John McGinn at Stamford Bridge and in launching an investigation to identify the culprit.
"Aston Villa will fully support that investigation. Both clubs have zero tolerance for the scourge of discrimination in any form when it occurs in our stadia."