Mr Allardyce brought defamation proceedings at the High Court in London against Kean, of Glasgow, who was Blackburn's manager during a 2011 pre-season Far East tour.
His solicitor, Hanna Basha, told Mr Justice Eady 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 over the internet and reported in the press.
She said Kean had acknowledged the allegation was completely untrue, withdrawn it, and agreed to pay a substantial sum in damages.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 distress caused and said he was pleased to set the record straight.
Ms Basha said Allardyce's claim against Blackburn had been settled and the club had published an apology on its website.