The 29-year-old has just 12 months left on his current contract with the club and has long been linked with a move away from the Gunners.
"I personally have had a great season but my goal has been to win trophies with the team and to bring the club back to its glory days," a statement from Van Persie read.
"Out of my huge respect for Mr Wenger, the players and the fans I don't want to go into any details, but unfortunately in the meeting it has again become clear to me that we, in many aspects, disagree on the way Arsenal FC should move forward.
"I've thought long and hard about it, but I have decided not to extend my contract."
Van Persie has been linked with the likes of Manchester United, Manchester City and Juventus in recent months.