Manchester City are interested in bringing the Dutchman to Eastlands this summer but Goal.com revealed that the Premier League champions will not offer the striker a contract above £198,000 a week due to his injury history.
|VAN THE MAN|
|7/1||Robin van Persie is 7/1 to be top scorer in the Premier League next season with bet365|
But Winterburn is adamant Wenger must continue to improve the squad in order to persuade Van Persie to extend his deal with Arsenal, which is set to expire next summer.
"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.
"Arsenal should not be a selling club. Forget about the board.
"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."
Winterburn admits his former club are struggling to compete with both Manchester clubs and claims Arsenal ambition to consistently secure Champions League qualification is not a sufficient achievement for a player.
"It's very difficult for Arsenal to compete with the likes of Manchester City, Manchester United and Chelsea," he said.
"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."