Walcott, 23, is into the final season of his current contract but has yet to negotiate an extension, with the two parties said to be at an impasse over wages.
"I still hope to sign him and the next two months will be vital because after that, the longer this situation lasts, then the more difficult it is," Wenger said. "Let's hope we can find a solution in the next two months."
There had been suggestions the England forward - who could sign a lucrative pre-contract agreement with another club in January - would be offloaded in the last transfer window to avoid him moving as a free agent next summer, which Wenger refuted.
However, the Frenchman acknowledged the situation was reaching a make-or-break point with the player he signed from Southampton as a 16-year-old.
Asked if Arsenal would consider selling Walcott during the January transfer window, Wenger added: "It will depend on the needs of the team and the club.
"We can afford to go to the end (of the contract) if needed."
Walcott has not started a match since the opening day against Sunderland and received a mixed reception from some sections of the Emirates Stadium when coming on against former club Southampton on Saturday.
Wenger added: "The fact he doesn't play regularly at the moment is right, but it's not necessarily linked with his contract situation. No player is happy when he doesn't play.
"At the moment, [Laurent] Koscielny doesn't play. He has just extended his contract and still doesn't play because the defence is doing well."