Walcott’s current deal is set to expire in July 2013, but talks over an extension have reportedly stalled as the England international looks for assurances that he will feature in a more central role in the future.
Following the arrivals of Lukas Podolski and Olivier Giroud in the summer, Walcott has seen such opportunities limited, but Wenger ruled out an exit for Walcott during the upcoming transfer window.
When asked if the player could leave in January if a contract extension is not agreed, Wenger told reporters: "No. Will he stay until the end of the season? Yes.
"If I listened to the media every day it gets more expensive. We continue in a positive way. We have until the end of December to sort it out.
"He can play both positions. He can play up front, and he can play up front with Giroud as well. He has found a good mixture on the wing. It makes him difficult to stop. He finished as a centre forward and he scored.
"With the midfield we have Jack Wilshere, Santi Cazorla and Mikel Arteta, in any position they will find you if you move well."
Walcott has impressed up front in recent performances, opening the scoring in the 2-2 draw with Schalke on November 6 and netting against rivals Tottenham on Saturday, and voiced his opinion that he had proven himself in the role following the 5-2 victory.