The goalkeeper spent last season on-loan with Celtic.
"He stays with us at the moment," Hasenhuttl said of Forster. “We have Alex McCarthy as our number one. I think Alex made a fantastic development in the last games.
“Then on the number two position, Fraser and Gunny, one will maybe leave us and the other one will stay with us.
“Especially with the ball, we have created some habits and some automatism that Fraser just has to learn. He definitely can do it because he is also open-minded and tries to improve his game.
“The goalkeeper game is changing all the time, it's developing, especially what you demand with the ball is definitely completely different now.
“Andrew (Sparkes, goalkeeping coach) does it over a year now with Alex and you can see that he's normally a little bit ahead of the other guys."