The lock forward suffered a knee injury during England's tour of New Zealand when he was leading the side against the Crusaders earlier this year.
Upon his return the 26-year-old, who signed a new two-year deal last November, underwent further investigations on the problem and Leicester director of rugby Richard Cockerill revealed that he had started his rehabilitation following surgery.
He told the club's official website: "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.
"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."
A statement on the club's website said that Slater "faces several months on the sidelines", with Cockerill telling the BBC "that's probably his season over".
Slater, who was included in England's squad for the Six Nations this year, has yet to make his full international debut.
Since joining Leicester in 2010 he has made 94 appearances and was part of the side that won the Premiership title in 2012-13.
Leicester began their season with a 36-17 victory over Newcastle Falcons on Saturday with Vereniki Goneva scoring a hat-trick of tries.