The 23-year-old's current contract with the Gunners runs out at the end of the season and a new deal has yet to be struck.
"No," Wenger told reporters when asked if the England international could leave in January. "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."
Wenger also praised Walcott's versatility and his playing style.
"He can play both positions," said the Frenchman. "He can play up front, and he can play up front with [Olivier] 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 scored in Arsenal's 5-2 victory over Tottenham Hotspur at the Emirates on Saturday.