Van Persie was carried off in the 15th minute of a friendly between the Netherlands and Italy after a tackle by Italy defender Giorgio Chiellini.
"The prognosis is that I will be out for four to six weeks," van Persie told the Dutch television station NOS.
Speculation is also rife that he could now be out for the remaining part of the 2009-10 season.
Van Persie, 26, has scored eight goals in his last 11 matches for the north London club who are second to Chelsea in the Premier League. Arsenal host Chelsea at the Emirates stadium on November 29.
'No wrong in return of R R Patil as home minister'
Liverpool rout Real, Chelsea eliminate Juve
No intention of seizing Pak's nuclear weapons: US
Wish Shah Rukh Khan a happy birthday!
Milan set to seal Adiyiah, Beckham deal