Wilshere has not featured for the Gunners in a competitive game this campaign after suffering the injury in a pre-season friendly against the New York Red Bulls at the Emirates Stadium.
A statement on the club's official website on Monday evening read: "Jack Wilshere has this morning undergone ankle surgery at a central London hospital.
"Under the care of two world-renowned surgeons, Jack has had the stress fracture of his right ankle repaired.
"The initial indications are that the surgery has been successful and Jack will now undergo a prolonged period of rehabilitation. It is envisaged that Jack's rehabilitation will take a period of four to five months."
Wilshere had initially expected that the problem would not require surgery but the club confirmed last week that an operation would be necessary.
Wilshere added on the official website: "I've had my operation this morning and all went well. My ankle is little bit sore, but I'm already looking forward to working hard to get fit and playing again.
"I would also just like to say thanks for the many messages of support I have received from you - the fantastic Arsenal supporters. See you all soon at the Emirates."
The Gunners' prediction means Wilshere may not play competitively for the club until late February.
There was further bad news earlier in the day when Captain Robin van Persie confirmed he was in no hurry to commit his long-term future to the club.
Van Persie scored his 100th Arsenal goal in Saturday's 3-0 victory over Bolton, and the continued good form of the new skipper could be the key to turning around the Gunners' poor start to the season.
The 28-year-old is understood to have sought assurances from boss Arsene Wenger that the club's future was still bright following the sales of mdfielders Cesc Fabregas and Samir Nasri,
who refused to extend his contract and was sold to Manchester City.
Wenger could be faced with a similar problem to the Nasri one if Van Persie does not sign an extension to his current contract, which expires in 2013, but the Dutchman insists talks are on the backburner for the time being.
"Maybe I'll leave it [holding immediate talks] because the main thing for me is the team and that we play well," Van Persie was reported as saying in a number of national newspapers.
"I do believe you have to look for a right moment to sit down and speak about your own stuff.
"It's not only about me, it's about the team. We are not talking now because we are so busy - we have games every three or four days.
"I don't think it's clever to do that during the season."
He added: "I still have almost two years left here, so for the moment that's fine. But I don't know, we will have to look.
"I'm happy with my contract, I'm good. That's how it is, I can't say much more. I can't look into the future."