Chelsea say they are "disgusted by the racial abuse directed towards Nicolas Jackson via social media" following their FA Cup semi-final defeat by Manchester City on Saturday.
Mauricio Pochettino's side lost 1-0 to City at Wembley in a game in which Jackson missed three promising chances.
"There is no place in society for any form of discrimination and we operate a zero-tolerance approach to any incident of this nature," said a Chelsea statement.
"The club will support any criminal prosecutions and take the strongest possible action, including bans, against any individual who is found to be a season ticket-holder or member.