The Gunners' eight goal frontman was stretchered off after 15 minutes of the Netherlands international friendly against Italy on Saturday and taken to hospital following a strong challenge from Giorgio Chiellini.
Netherlands coach Bert van Marwijk suggested after the match in Pescara that van Persie had torn ligaments in his right ankle.
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News of van Persie's injury will come as a blow to Gunners boss Arsene Wenger.
Wenger recently admitted that he was keeping his fingers crossed that the latest round of international games did not result in a spate of injury problems.
"You can have two or three difficult games in November," said the Frenchman.
"You have decisive games in the Champions League and the players go on international duty.
"Now we have that break and all the players fly away so let's hope they all fly back in good shape.
"After that we go to Sunderland and that is always a difficult place to go."