The France youth international, who joined the Gunners from Lens last year, suffered the problem after returning to pre-season training.
He will now miss the club’s involvement in the Durban International Cup in South Africa, whilst he is also set to be absent for the start of the U18 campaign.
Although he joined the club as a striker, Fortune was soon moved out to the left wing. He provided some tantalising glimpses of his potential in his first season but lacked ruthlessness in front of goal.
His absence could see new first-year scholar Emile Smith-Rowe push into the U21 side during pre-season.