The 26-year-old, who Wenger signed from Southampton as a 16-year-old, is out of contract at the end of next season, when he could leave for nothing.
Negotiations have begun over a new deal, giving Wenger cause for optimism that Walcott, who was the subject of protracted talks before signing his current deal, wishes to stay.
Walcott's present deal was signed in January 2013, fewer than six months before that contract was due to expire.
Wenger hinted Arsenal would hold on to the England forward this summer and would only consider selling Walcott in the January 2016 winter transfer window if no new deal is agreed.
"We are already talking to him," said Wenger, speaking ahead of Monday's home clash with Swansea.
"When you go into negotiations with a player it is with a desire for him to stay. He looks keen to do it and I am keen to do it, so let's see what comes out.
"We are always in the mode where we want to be successful. In case of failure, we'll see.
"But in case of failure - that will only be in December next year."
Carl Jenkinson is another player whose future is to be decided after he has impressed on loan at West Ham.
"I met Carl this week to speak about his future. We will sit down together at the end of the season," said Wenger, who must decide if he will loan Jenkinson out again next term.
"My target is to keep him here at the club. It's important that, if he comes back, he has a good chance to play."
Arsenal have won nine of their last 10 games - the only dropped points a draw with champions Chelsea - in a run Wenger hopes to continue against Swansea on Monday night.
The Frenchman, whose team defend their FA Cup crown against Aston Villa in this season's final on May 30, is determined to finish the current campaign well before building a title-challenging team for next term.
"It is difficult to know how strong every team will be next year but you gain as well your confidence from what you have achieved the year before and how you have moved forward," Wenger added.
"It is important to finish as close as possible to Chelsea and come out of the season and think we are ready to challenge next season."
Manchester United on Thursday signed Holland forward Memphis Depay for £25million.
Asked the importance of doing transfer business early in the summer, Wenger said: "What you want to do more than do it early is to do it right.
"You have to take the opportunities when they come. Fix your target, be on alert and do it when it's possible."
source:eurosport