Arsenal's highest-paid player, who earns about £90,000 a week, was carried off with damaged ankle ligaments during a friendly between Italy and Holland last weekend.
Van Persie's absence could wreck the Gunners' season - and will widen the rift between Europe's superclubs and the national FAs over the use of their star players.
Marloes van der Lan, spokeswoman for the Dutch FA, told the Sunday Mirror: "He falls back into the insurance of his club. The federation is not paying out money for the injured player.
"We do compensate Dutch clubs who have players injured with us. But in the case of Van Persie, Arsenal should perhaps go to the English FA."