The England forward's current deal runs out in the summer, and the club have been in negotiations to close out terms on a lucrative extension which would make Walcott among the highest-paid players at the club with a five-year contract worth around ?90,000-a-week, boosted by lucrative add-ons.
While not taking anything for granted, Wenger hopes a positive end to the saga can be reached in the coming weeks, and he said: "We are on a good road but in this situation as long as nothing is signed, you are cautious, but I am optimistic we will get to a happy conclusion."
He added: "I was optimistic last week, but I am a bit more optimistic this week. Discussions progress smoothly, but slowly."
Asked whether he expected a deal to be signed by the end of January, Wenger said: "Yes."
It had been suggested part of any new deal was subject to Arsenal meeting Walcott's desire to be deployed more in his preferred role as a central striker.
He has revelled in the position during recent weeks, crashing in a hat-trick in the 7-3 win over Newcastle. Wenger, though, insisted there was no such clause, and that he would never agree to any tactical demands from any player.
"Theo can have a preference maybe to play through the middle, but first of all he never made a condition of that to extend his contract, because certainly I would not accept it," the Arsenal manager said.
"Secondly he was always happier when he plays, whether it is on the flank or through the middle. That has no influence at all.
"Theo just wants to play and he is happy to play on the flank as well."