This has made the Gunners fear the Dutchman, 28, could be the next big name to walk away.
Fab, who left London to join boyhood idols Barcelona in the summer, said: "I don't know if they will keep him. It is his decision.
"At his age, if he wants to move it will be his last chance and if he wants to stay he will stay and retire at Arsenal."
Van Persie has just 18 months left on his current deal with the Gunners.
Fabregas said: "We will see what he decides but he's been very loyal to Arsenal for a few years. He is a key player, the fans love him.
"He is the role model of Arsenal, the star player and they cannot afford to lose him."