"Arsenal have bought a couple of players and you can only hope they continue to strengthen throughout the summer," Winterburn told talkSPORT.
"Somehow that might persuade Van Persie this is the right club and that they've started to move in a different direction from the past five or six years."
Winterburn fears the Gunners' descent into a "selling club" has rendered them incapable of challenging for the title.
"Arsenal should not be a selling club. Forget about the board," he said.
"The most important thing is the manager and players and that the players go into each season believing they've got a chance of winning something.
"The precedent was set last year losing Fabregas and Nasri and now it looks like Van Persie may go, and that starts to put doubt into players' minds.
"As a player you need to believe you can challenge for the title and last year Arsenal were miles away.
"Getting into the Champions League is great, but as a player that's not enough. That's the minimum requirement.
"As a player you want to be in the mix [for the title] in March and April when the excitement really starts."