Davis told BBC Stoke & Staffordshire: "I'll be speaking to [assistant boss] Steve Bould at Arsenal to see if we can keep it going for another month.
"He feels comfortable in this company, he's enjoying it and wants to stay."
Once he has completed his three-game suspension at MK Dons on Saturday, Aneke is due to return to action for next Tuesday night's home game against Oldham, after which his future is open-ended, assuming the Gunners are happy for him to stay on.
"There is no timescale on youth loans," said Davis. "They can stop as long as we want to have him and he wants to stay.
"And what I've seen so far suggests that he's going to be a good addition who can only get better... as long as he stays on the pitch."