With the England wide-man's current deal set to expire next summer, the Gunners are believed to have made an offer worth €95,000 per week.
Walcott has played only a bit-part role so far this season, and the Frenchman has admitted he hopes to have his contract situation resolved quickly.
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"I still hope to sign him and the next two months will be vital because, after that, the longer this situation lasts the more difficult it is.
"Let's hope we can find a solution in the next two months but the fact he doesn't play regularly at the moment is right, but it's not necessarily linked with his contract situation."
After a number of infamous contract stand-offs with key players such as Robin van Persie and Samir Nasri in recent years, Wenger has seemingly taken a stricter approach to the dealing of Walcott.
However, Chelsea and Manchester City are reportedly interested in the Gunners man, whilst Liverpool were strongly linked with a move for him during the summer.
"It will depend on the needs of the team and the club [as to whether Arsenal will sell in January]," said the Frenchman. "We can afford to go to the end [of the contract] if needed.
"No player is happy when he doesn't play."