The Dutchman has a contract at Arsenal until 2013 and speculation is rife that he could leave the club.
Arsenal have got themselves into a similar position to the one they had with Samir Nasri as they look to keep Van Persie at the club.
Nasri also had a year left on his deal last summer and despite promising to keep him at the club, Wenger failed and the Frenchmen went to Manchester City.
Kolo Toure, Emmanuel Adebayor, Gael Clichy and Nasri have all left Arsenal for City and there are fears among some sections of the club's support that Van Persie will go the same way.
But Wenger says he is confident he can keep the 28-year-old at Arsenal - and he will strive to do so before Euro 2012 kicks off on June 8.
Asked whether he will try and get Van Persie to commit before the start of this summer's tournament, Wenger said: "That's what I will try to achieve.
I am confident for two reasons. First of all I want to convince him to stay here and to extend his contract. Secondly he is not free at the end of this season, he has one year of his contract left anyway.
"He has grown mentally and you see it better when you look back with a distance. He has grown into a leader and he is a great captain now, on and off the pitch."