Police are investigating the allegations from Euell, 32, which have been supported by Blackpool manager Ian Holloway.
A supporter who was detained on the night of the match after being ejected by stewards has been released on bail.
Stoke chief executive Tony Scholes said the club have it in their power to take firm action, which could mean a ban from the club's stadium - depending on the outcome of the police investigation.
Scholes said: "On behalf of Stoke City Football Club, I would like to apologise to Jason Euell, and indeed Blackpool Football Club, for the behaviour of this supporter."
He explained: "We will take a hard line because we do not tolerate racist or abusive language of this type. We must first wait though for the outcome of the police investigation into this incident before imposing our sanctions upon this person.
"The Britannia Stadium is a family environment and it is staggering that in this day and age, people may be subjected to this kind of language."