The 23-year-old, who is out of contract at the end of the season, has yet to commit himself to a new Arsenal deal.
Walcott made the most of a rare start this term with a hat-trick in Arsenal's dramatic midweek 7-5 League Cup win over Premier League rivals Reading.
If the player and the Gunners do not come to terms before January, Walcott will be able to negotiate a pre-contract agreement with a foreign club or leave the Emirates on a Bosman free transfer at the end of the season.
Wenger, asked if Arsenal would have to sell Walcott if no agreement has been reached by the start of the new year, said Thursday: "I would not like to. I have not even thought about that, because at the moment I think we will still manage to make a deal with him."
The Frenchman added: "There is urgency (on talks).
"We want to sort it out before Christmas, one way or the other."
Wenger urged Walcott to be patient, saying the former Southampton player would get more chances to star in his preferred central attacking position, as he did against Reading, rather than on the wing.
"I always said it is important that he plays in the right position and that certainly would be through the middle in the future, he showed that again on Tuesday night," Wenger said.
"The positions on the flanks are changing, you need to do a lot of defending. He should be dedicated more to offensive work."
Wenger added: "I have always come out publicly and said that I see him in the future through the middle as well. He loves to score goals, Theo.
"He has good pace, (makes) excellent, intelligent runs, and what has changed with Theo is he's (become) a very good finisher."
Source: AFP
Source: AFP