Van Persie was carried off in the Oranje's friendly against Italy on Saturday night after a heavy challenge from the Azzurri's Giorgio Chiellini.
Van Persie told Dutch television programme Studio Voetbal: "I have had a scan.
"I am probably out for between four and six weeks."
He did, however, absolve Juventus defender Chiellini of blame, adding: "There was no intent."
The 26-year-old is a key player for Gunners boss Arsene Wenger and his absence could seriously harm the club's Premier League title bid.
Van Persie lasted just 10 minutes of Saturday night's game in Pescara with Holland boss Bert van Marwijk revealing afterwards: "The first diagnosis is a bad ankle injury.
"It's terrible for Robin and is not good for Holland or his club."