It is a crippling blow for boss Wenger's Premier League and Euro hopes.
Van Persie was left in agony 10 minutes into Holland's goalless friendly with Italy on Saturday after a challenge from Giorgio Chiellini.
He has scored seven league goals so far this season, leading the line for the in-form Gunners.
Van Persie, 26, was in Amsterdam last night undergoing scans as medics looked to rule out any fracture to the joint, The Sun reports.
They have confirmed the ligaments are torn.
Van Persie remains optimistic.
He said: "I am probably out for between four to six weeks."
Wenger said: "The footage doesn't look that bad, but the seriousness of the injury does seem to be."
Arsenal are second in the table and Wenger will be hoping the loss of the Dutchman does not mirror striker Eduardo's situation from two seasons ago.
In February 2008 the Croatian broke his leg - and his absence led to them surrendering a place at the top of the table, as they finished third. (ANI)