Walcott has been on the books of Arsenal since 2006 and has a contract until June 2019.
The England international has suffered with injury problems in recent years, and he is not guaranteed a place in the Gunners' starting lineup.
In fact, the 27-year-old, who can play as a forward or as a winger, has not started a Premier League match for the North London outfit since February 28.
Wenger has admitted that he does not know if the former Southampton attacker will remain at Arsenal beyond the summer.
"I don't know yet,” the Frenchman told Sky Sports when asked if Walcott could leave Arsenal. "I'm not in transfer mode at the moment, but I think he has gone through a bad spell.
“He has responded very well, because he said: 'Okay, I will work harder and more.’ Even when he comes on now, you see he is motivated and focused. Let's see how well he finishes the season now.”
Walcott has scored five goals and created 17 chances in 26 Premier League appearances for Arsenal so far this season.
The Englishman has a pass accuracy of 80% and a shot accuracy of 61%, and has won five headers, 23 dribbles and seven tackles.
Arsenal are fourth in the Premier League table at the moment with 63 points from 34 matches.