Dein was a key boardroom figure at the club during a time when they fought summer-on-summer to keep the likes of Patrick Vieira and Thierry Henry.
The Gunners say they expect van Persie to honour the remaining year of his contract, even though he will not sign a new one.
And Dein told Sky Sports News that a sale must happen soon or not at all.
He said: "In my experience there is one or two clubs in the background, and they're going to come out and make an offer, and then it's up to the club to decide what they're going to do.
"All I would hope for the stability of the club is that, whether he stays or goes, it happens quickly. You don't want a repeat of what happened last year, when you had Samir Nasri and Cesc Fabregas stringing it out.
"Clubs this week are going back for pre-season training and you want a settled squad."
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