The 19-year-old is due to return to his parent club at the beginning of October, but Davis is hopeful that an agreement can be made.
"Chuks has worked very hard. He has to get used to working hard here and he is getting fitter all the time. I would imagine he doesn't do too much running without the ball at Arsenal but he has to do that here," Davis told the club's official website.
"I am sure he feels comfortable here and is enjoying his stay here and he wants to extend it. I will speak to chairman and see if we can keep it going for another month. We can have him as long as they will allow it and as long as we want him to stay.
"We want him to stay. We need to speak to Steve Bould and those at Arsenal to see if they are happy. If they are then we will do it. What we saw at Stevenage was enough to suggest he is a good addition."
Aneke has made one first-team appearance for Arsenal.