Kean had claimed that Allardyce was sacked from his previous post at Rovers because he was a "crook".
Allardyce brought defamation proceedings in London's High Court against Kean, who was Blackburn's manager during a summer 2011 pre-season tour of the Far East.
His solicitor, Hanna Basha, told Mr Justice Eady that the "false and defamatory" allegation, made while Kean was drinking in a bar in Hong Kong, was filmed on a camera phone and subsequently published to many hundreds of thousands of people over the internet and reported in the national press.
She said that Kean had acknowledged the allegation was completely untrue, withdrawn it, and agreed to pay a substantial sum in damages.
Kean's counsel, John Mehrzad, offered his apologies for the hurt and distress caused and said he was pleased to set the record straight.
Ms Basha said that Allardyce's claim against Blackburn had been settled and as the club had published an apology on its website, it was not a party to Wednesday's statement in court.
