Wilshere picked up the injury in the 2-1 defeat to Manchester United at the weekend.
The midfielder sustained the injury in the 2-1 defeat to Manchester United in the Premier League at the Emirates Stadium on Saturday.
A statement on Arsenal.com, read: "Jack will now commence his injury rehabilitation programme with the Arsenal medical team.
"Everyone at the club wishes Jack a speedy recovery and look forward to seeing him back on the pitch as soon as possible."
It could prove to be a big blow for Arsenal who have as many as nine first team players on the sidelines, with fellow central midfielder Mikel Arteta also in the Gunners' treatment room.
As a result of his injury, Wilshere will be out until late February, meaning he could miss at least one leg of the Champions League second round tie - which his side managed to secure qualification for last night with a 2-0 win over Borussia Dortmund.
Some of the higher profile games the 22-year-old is likely to miss include games against Liverpool, Southampton, Manchester City and Tottenham - games which could prove to be key to Arsenal's chances of qualifying for the Champions League once again.