The England international's contract expires next summer but the Gunners are yet to tie the attacker down to a long-term deal at the Emirates.
"My desire is to keep Theo. We do what's needed to keep him and hopefully, [as] I told you many times, we can sort this situation out very soon," Wenger told the club's official website.
"Very soon is before the end of December - one way or another, that is for sure."
Walcott has been in good form this season and is the the club's leading marksman, scoring eight goals in all competitions, as he pushes to be used more regularly as a center forward.
"I don't want to go into any details but you can believe me [that] we do the maximum we can to keep our best players," Wenger said.