The Dutchman has been vital to the north London side's turnaround this season, as they overcame as sluggish start to storm into third place in the Premier League, but has just over a year remaining on his contract.
Petit believes Arsenal must qualify for the Champions League, as well as prove to Van Persie that they have the potential to win trophies in the next few seasons, in order to keep him.
By the conclusion of this campaign it will have been over seven years since Arsenal last won a trophy and, at 28, Van Persie is beginning to edge towards the latter stage of his career.
The prolific forward has scored 33 goals in all competitions this season but has only won a solitary FA Cup since joining Arsenal in 2004.
Petit said: “My main worries will be that I hope Arsenal won't let him go at the end of the season because he is a class player, an international player.
“Since he's far from the injuries he can show his full potential. He is magnificent. That's why I pray for Arsenal fans and for Arsenal that he will stay at Arsenal.
“But It depends on qualification of the Champions League. It depends as well on the ambition that Arsene will show to Van Persie to stay at the club.
“For Van Persie, his talent and what he has shown this season it will be a shame for him if he cannot win a single title in his career.
“That's why I think he must think about it very carefully.”
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