The 26-year-old, whom Wenger signed from Southampton as a 17-year-old, is out of contract at the end of next season and negotiations have begun over a new deal.
The Frenchman is optimistic the fact discussions have started means Walcott wishes to stay.
“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.”