The four-minute clip, which appeared on YouTube and on the website of a Rovers fan group named BRFC Action Group, was apparently filmed during a tour of the Far East last summer.
As well as the remarks about Allardyce, the video also appears to show Kean predicting Rovers will finish in the top half of the Barclays Premier League table and win the Carling Cup. Rovers finished second-bottom and were relegated to the Championship.
Attempts made by Press Association Sport to contact Rovers were unsuccessful.
Kean and Blackburn owners Venky's were this season the targets of sustained protests from the Rovers fans unhappy at the running of the club.
Kean was appointed manager as a replacement for Allardyce, who was sacked in December 2010.
Allardyce has since led West Ham to the final of the Championship play-offs.
Rovers officials are expected to hold talks with Kean in the coming days, but have so far shown no signs that their loyalty to their manager is running out.