The Dutch striker announced his intention to not extend his current deal with Arsene Wenger’s side beyond the end of the upcoming campaign and, most recently, pulled out of the club’s pre-season tour of Asia to reportedly try and force a transfer away from the Emirates.
Graham feels that a quick solution would be best for both parties, allowing Arsenal to move on and avoid a similar hangover after selling Cesc Fabregas and Samir Nasri in the early part of last season, which resulted in a disastrous start to the campaign.
He told ESPN: “You have to sell Van Persie and move on, and Arsenal need to tell everybody that they can’t compete on the financial aspect, that they can’t compete with Chelsea, Manchester City, even Manchester United.
“If the club have to sell their top player they also have to tell their fans it has to be done this way. Last year it dragged on with Fabregas and players were brought in late and Arsenal had a bad start, culminating with an eight-goal hammering at Old Trafford.
“With Van Persie it needs to be done and dusted - he has made his statement, he has said the club lack ambition so the club need to get the best offer they can and move on. Van Persie has another year left on his contact and Arsenal tried to negotiate.
“He knows he can get a better offer elsewhere but it’s not just about an offer, it is about winning trophies and Arsenal haven't won a trophy for five or six years and that comes into the equation, whether the club are ambitious enough.”
However, the 67-year-old does have sympathy for Wenger and the Arsenal hierarchy, adding that he is unsure of how he would have handled the situation if he were still in charge.
“What more could Arsenal have done with Van Persie? I am not sure they could have done more, or if I would have done it any differently," he concluded.