Walcott's current deal expires in the summer, and as yet there has been no positive conclusion as negotiations continue with the parties still some distance apart over wages.
Wenger has always stressed the club's desire to keep the player, and said: "There is urgency, how much I don't know, but there is urgency. We want to sort it out before Christmas, one way or the other."
Should an agreement not be reached by January, Walcott would be free to open negotiations with overseas clubs to potentially sign a lucrative pre-contract agreement on a Bosman transfer.
Asked by Press Association Sport if Arsenal would have to sell Walcott if no deal was signed by the start of 2013, Wenger said: "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."
Another issue of contention with Walcott is understood to be his desire to play as a central striker, rather than being utilised down the flank.
Wenger feels Walcott will eventually be deployed in his preferred attacking role, as he was to devastating impact at Reading on Tuesday night, but called for patience.
"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: PA
Source: PA