Campbell, who formed part of the back-line during the club's Invincibles era, when they became the first team in Premier League history to go an entire season unbeaten, believes the 28-year-old is fed up with each season delivering nothing.
Van Persie this week questioned the club's transfer policy and ambition in a message to fans on his personal website in which he confirmed that he would not be extending his contract at the Emirates.
His announcement prompted Alisher Usmanov, who with his business partner Farhad Moshiri is Arsenal's second largest shareholder with a 29 per cent stake, to issue an open letter to the board criticising the direction in which they are taking the club.
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The 37-year-old told the Daily Mirror: “I can understand why Van Persie has decided not to sign a new contract.
“When you go back to your club for pre-season, and you look around the changing room, you want to see top players around you and think, ‘Yeah, we’re going to do something this season. This is where it’s happening.’
“That’s what Van Persie is looking for. And maybe he’s had a look and thought, ‘Is this it?’
"Maybe it will be different in a couple of weeks, maybe there will be a cast of new faces to make him feel different.
“Players come and go. It’s a fact of life in football."Van Persie joined Arsenal in 2004, just weeks after they celebrated winning the title in an unbeaten season, and Campbell believes that while the trophies may have enticed him to the club, the fact that they have dried up has now convinced the Dutchman his future lies elsewhere.
He added: “When he signed for Arsenal, he was surrounded by top-quality players, big characters, and it was fantastic. Inside his head, he knew it was the place to be - alongside Thierry Henry, Dennis Bergkamp, Patrick Vieira, myself, Tony Adams, Jens Lehmann, Ashley Cole.
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“Van Persie came in, mixed it with a team who were used to winning things, and he got the taste for it. He joined The Invincibles and won the FA Cup in his first season, but since then... nothing.
“Robin has seen the good times, and been part of the good times, and he wants more of it.
“Basically, his statement is shorthand for saying his patience has run out, and that he can’t wait for ever if he’s going to win major trophies in his career."
Campbell warned fans, though, that a change in manager is not the way forward and feels Arsene Wenger is one of the only managers who can address this slumo the club finds itself in.
That said, Campbell acknowledges that the club must be more ruthless in its transfer policy. He believes it will only be spending on star players rather than healthy receipts that will convince Van Persie to change his mind and opt to stay at the Emirates.
“Arsenal have established this model of financial prudence, they have a fantastic new stadium, they are bringing the club’s debt down ahead of schedule and everything is tickety-boo off the pitch," said Campbell.
“They are making money, they are in good shape and everything is well-run... but they are not winning. The real sting is in the punchline.
“Changing the manager isn’t going to make any difference. If there are any Arsenal fans who are tempted to call for Arsene Wenger to step down, is a new manager going to get any more firepower in the transfer market?
“Unless someone produces £100m and Wenger brings some big signings to the club, I can’t see Van Persie changing his mind.