Aneke's Arsenal deal expires in June.
Davis told The Sentinel: "It's something that I've thought about. It maybe a bit early for him to decide what he's going to do in the summer but I know his agent is very good.
"Chuks has spent most his development here and (his agent) will see the change over the period he has been with us, taking bigger strides than if he had stayed playing in Arsenal's under-21s or gone to a club that wouldn't have helped him as much as we have.
"We have done everything we can to help him as a player and that shows in his performances. Maybe it is something that will develop over the next few months."