Giorgio Chiellini's challenge left Van Persie rolling around in agony but, after being examined by medical staff at the Adriatico Stadium, he flew home with the Netherlands squad and spent the night with his family in Rotterdam.
A Dutch Football Association (KNVB) official told BBC Sport that Van Persie had a partially ruptured ligament.
"It's a serious injury," the KNVB official said, following concerns that Van Persie had broken his ankle. "We don't know how long it will take him to recover from it.
"It's impossible to put an exact timeframe on it because some players heal more quickly than others. It's not a short-term injury."
Van Persie's career in north London has been interrupted by a succession of injuries - he broke toe in January 2006, fractured a metatarsal in January 2007 and damaged a knee on international duty in October 2007.
Van Persie has been a key player for Arsenal this season
After enjoying a relatively injury-free 2008-09 season, he has started all of Arsenal's Premier League games this season, is their second-highest scorer with eight goals and also has eight assists.
All of Van Persie's goals have come in the last 11 games and the former Feyenoord forward has helped propel Arsenal to within five points of Premier League leaders Chelsea with a game in hand.
His absence leaves the Gunners short in attack, with Nicklas Bendtner sidelined until mid-December following minor groin surgery and Theo Walcott nursing a knee ligament problem.
Arsenal are also currently without the injured Gael Clichy and Denilson.
Any lengthy absence for Van Persie will also be felt by the Dutch as they prepare for the 2010 World Cup.
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