FA spokesman Adrian Bevington said: "We take very seriously any individual running onto the field to confront players."
He added: "We will be discussing the situation with both clubs, the officials and the police and we hope that not only is the individual banned by the club but also goes through the legal system and is subjected to a banning order. There is no place for such behaviour in football."
Stoke boss Johan Boskamp said: "What happened was a disgrace and I can only apologise."