Wenger's position as manager has been called into question following the defeat, and Winterburn believes that the Frenchman could consider his options during the summer.
"I expected Arsene Wenger to sign his contract a few weeks ago," Winterburn told BBC Radio Five Live.
"He said he did not want to disturb the team. I'm now starting to think if Arsenal do finish outside the top four, Arsene might be thinking it's time to leave.
"Personally I hope not. I hope he changes his ways and strengthens the team in the summer. I don't think they are far away but it's been a disappointing end to a league game campaign for them."
Arsenal face Wigan Athletic in the semi-finals of the FA Cup at Wembley on Saturday.