During Rovers' 2011 pre-season tour of the Far East, Kean was filmed talking to fans and making allegations about Allardyce. The video was posted on YouTube. Kean served under Allardyce as first-team coach before the taking over in the wake of the 58-year-old's surprise sacking in December 2010.
Allardyce initiated legal proceedings against Kean and Blackburn over the remarks and reached an out-of-court settlement last week. On Tuesday, Rovers issued a statement on their official website under the headline "Sam Allardyce: An apology".
It read: "During the summer of 2011, Mr Steve Kean was this club's manager and accompanied the players of Blackburn Rovers on a pre-season tour of the Far East."
The statement went on to say that Kean was filmed in a bar making false allegations about Allardyce, adding: "We apologise to Mr Allardyce for the embarrassment and distress caused to him by Mr Kean."