Goal.com exclusively revealed in October that contract talks had reached a stalemate, before a breakthrough was made in late November as Walcott lowered his wage demands.
The attacker's contract expires in the summer, leaving him subject to reported interest from a number of Premier League rivals, including Liverpool, Manchester United and Manchester City.
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"I'm convinced it will have a positive effect on the rest of the team if we sign him," Wenger told reporters on Thursday. "We want to keep him here.
"Our priority is to keep players at the club. We have rebuilt a team and we have to show that we can be consistent now."
Wenger also confirmed that Johan Djourou and Sebastien Squillaci could be leaving the Emirates Stadium this month, adding: "If Squillaci finds a club, he might go. Djourou might go as well."