Oyston is alleged to have made the remark as part of a series of abusive messages between himself and the fan.
Blackpool were not available for comment on whether Oyston sent the messages, which have been widely reported over the last 24 hours.
The Football Association has said it is aware of the row.
Oyston, a member of the Football League Board, is also alleged to have told the supporter to "enjoy the rest of your special needs day out".
Blackpool Supporters Trust has issued a statement condemning the alleged texts from Oyston and called for an explanation.
The statement added: "Regardless of the context of the conversation, these are shocking sentiments to read from the chairman of a football club addressing a supporter.
"The language is wholly inappropriate from someone in a position of responsibility towards his customers and the community that Blackpool FC serves."
Oyston's weekly column in the Blackpool Gazette has been dropped by the newspaper.
Blackpool are currently bottom of the Championship and lost 6-1 at home to Bournemouth on Saturday.