The Ivory Coast international, who was troubled by the groin problem during Chelsea's Premier League and FA Cup double-winning campaign last season, successfully had "minimal invasive surgery" on Friday, the club said on their website on Saturday.
The 32-year-old is expected to return to full training in three weeks meaning he will miss the August 8 Wembley clash with United and possibly the beginning of their league title defence at home to West Bromwich Albion on August 14.