Walcott's current deal runs out next summer and the England international is reportedly unhappy with his lack of starts so far this season.
Wenger is keen to keep the 23-year-old but claimed that, one way or another, the saga will be sorted out by the end of the year.
"My desire is to keep Theo. We do what's needed to keep him and hopefully [as] I told you many times, we can sort this situation out very soon - very soon is before the end of December, one way or another, that is for sure," said Wenger.
"I don't want to go into any details [over wage negotiations], but you can believe me [that] we do the maximum we can to keep our best players."