The 26-year-old suffered the injury earlier this summer during England's tour of New Zealand.
"It's very disappointing for us and obviously for Ed because he was in such good form last season and has become an important member of our squad," Leicester director of rugby Richard Cockerill told the club's official website.
"Ed's performances for Tigers have pushed his way into the England squad and I was very happy to be able to appoint him as Tigers club captain going into the new season.
"He is in good hands with our medical team, he's started his rehab from the surgery and we look forward to his return to training and playing in due course."
Leicester started their new Aviva Premiership campaign last Saturday with a 36-17 win at home to Newcastle Falcons.