The Gunners have suffered a nightmare week that has left their title challenge in tatters, going down to a humiliating 6-0 defeat to Chelsea at Stamford Bridge, and conceding a last minute equaliser in the 2-2 draw with Swansea on Tuesday.
22-year-old Wilshere has struggled for form and fitness this season, and was criticised by former Manchester United midfielder Paul Scholes this week who claims he has not improved since the age of 17.
And Robson believes Wenger is to blame for not bringing the best out of the England international.
“Jack Wilshere can be a world class player,” Robson told talkSPORT.
“Paul Scholes is right though, he hasn’t developed as well as everybody would have liked him to.
“He lacks discipline. He wants to get on the ball and charge into tackles, there seems to be no discipline.
“Who does that come from? The manager. Wenger has to make sure he has a role in the side where he knows what to do when he does, and doesn’t, have the ball.
“He can be petulant at times. Why is he petulant? Because the manager is petulant on the sidelines and it’s never been sorted out.”
Wilshere has been linked with a move to Barcelona, and Robson believes a move away from the Emirates could save his career.
“Jack Wilshere, at some point, may have to move away from Arsenal,” he added.
“If Arsenal change the manager and someone comes in who’s really disciplined and can get the best out of Wilshere, then Arsenal will still be the right club.
“At the moment though, with Wenger in charge, I’m not sure Arsenal is the right club for him.”