Kean, who took over after Allardyce was sacked in December 2010, appears in a clip posted by a Rovers fan group named BRFC Action Group which was apparently filmed during a tour of the Far East last summer and without his knowledge.
The clip shows Kean making disparaging remarks about Allardyce, who is now in charge at Championship club West Ham United.
According to the Telegraph, Allardyce has instructed his lawyers to begin proceedings against the man who served as his assistant at Ewood Park, and is also considering doing likewise against Blackburn's owners, Indian poultry producers Venky's.
Kean and Venky's were this season the targets of sustained protests from the Rovers fans unhappy at the running of the club, which has culminated in the club's relegation.
Allardyce has since led West Ham to the final of the Championship play-offs, and will return to the Premier League with the Hammers if they beat Blackpool at Wembley on Saturday.
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.