The Potters parted ways with Paul Lambert following his failure to keep them in the Premier League. And they are now aiming to appoint a new manager as they look to return to the top flight at the soonest possible opportunity.
Rowett is understood to be their chief target for the job, although he only signed a three-and-a-half year deal in January.
"Derby County Football Club can confirm that Gary Rowett has asked for permission to speak with Stoke City regarding the vacant manager's position," the club wrote in a statement posted on their official website.
"The club is now in discussion with Stoke regarding the matter and will update our supporters in due course."
According to Sky Sports, Stoke are willing to pay the managers £1.8m compensation clause and will hold further talks with the Rams over the potential switch.