Initial reports had suggested that the 19-year-old midfielder could make a recovery before Christmas, but the Gunners have denied those claims.
"Under the care of two world renowned surgeons, Jack has had the stress fracture of his right ankle repaired," read a statement on the club's official website.
"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 earlier revealed that the operation had been a success.