"It is not the news we wanted," Stoke's director of football John Rudge told the club's official website.
"We have already spoken to James' agent to see if there is any room for manoeuvre, but as things stand now he is out of contract and has rejected our new offer by letter.
"With him being out of contract it means he is very much in the driving seat when it comes to looking around elsewhere and weighing up other offers.
"At the moment it seems unlikely he will re-sign for us but we will still be trying to persuade him to think again because he is a highly-regarded young player who has been very consistent in recent seasons and at times outstanding.
"Tony Pulis is desperately keen to keep him."