Contract talks between player and club have stalled with the Gunners unwilling to meet Walcott's wage expectations, while his demand to play in a central role is also a sticking point.
He is out of contract at the end of the season and Wenger said in his press conference on Thursday that Walcott's future was now a matter of "urgency".
"I want it resolved by Christmas," Wenger said. "His right position is through the middle in the future."
The 23-year-old underlined his goalscoring threat by firing a hat-trick in Tuesday night's thrilling 7-5 Capital One Cup victory over Reading, and Wenger said: "Theo was superb last night. I always said that he will be a striker, so we are on the same wavelength there."
He added: "First of all he is a good finisher now. Secondly, he knows where to be in the box on the rebound.
"That's a quality of a striker you can't give, you feel it or you don't.
"He has the qualities to play as a striker and I hope he will be a striker at Arsenal. I always said I want to keep him.
"During the contract talks you have to give him credit for the way he's committed to the team and the way he behaves."