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: "On behalf of Stoke City, I would like to apologise to Jason Euell, and indeed Blackpool, for the behaviour of this supporter."
He added: "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 permanent 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."
Under their ground regulations, Stoke have it within their power to impose a ban of an indefinite length on the supporter if they feel it is appropriate.