The action comes in relation to unguarded comments made by Kean about Allardyce to supporters while secretly being filmed last summer.
The four-minute video, apparently filmed during Rovers' pre-season trip to Hong Kong, was released on the internet overnight.
In the video, Kean makes comments which appear to be critical of Allardyce during his conversation with fans in a bar.
In footage in which expletives are used, Kean also appears to predict Rovers will finish in the top half of the Premier League and win the Carling Cup.
Rovers were relegated from the Premier League amid scenes of unrest and protest against Kean and India-based owners Venky's.
Kean has been a hugely unpopular appointment since being promoted from within to replace the sacked Allardyce in December 2010.
Despite repeated calls for his removal by fans, Venky's have strongly backed him and are due to meet the Scot for post-season talks in India this week. A petition demanding Kean's dismissal has attracted around 4,500 signatures.
The BRFC Action Group, who have claimed responsibility for the footage, have long campaigned against Kean and Venky's.
They appear to have stepped up their activity since demotion to the Championship was confirmed and they have now arranged meetings with Government and the Premier League to air their concerns.
Blackburn MP Jack Straw has facilitated a meeting of the fans with sports minister Hugh Robertson next month and the Premier League have agreed in principle to meet them too.
A statement from the group's committee read: "We as a group of supporters, elected by its members, are disgusted that our club has been allowed to be pillaged with very little resistance from the regulatory bodies who should act in the best interest of the sport and supporters.
"We are pleased to announce that the Premier League has invited the BRFC Action Group for a full meeting to take place in the next couple of weeks.
"We are further delighted that the Sports Minister has also agreed to a similar meeting to discuss the goings on, behind the scenes, at our club."
Many fans pinpoint the club's takeover by Venky's as the start of their demise.
The action group do not believe the firm are fit to run the club.
Chairman Glen Mullan said: "This will give us an opportunity to put our case to the highest authorities and offer supporting evidence to why we feel that our club's activities have not been monitored at all, since the FA granted Venky's [status as] fit and proper owners.
"This will include agents' roles during their ill-fated tenure and lack of administration structure at the club."
Venky's said last week they had no intention of relinquishing the club and have never given any indication they would like to replace Kean.
Indeed, at times it has appeared that the only contact the club has with its distant owners is through Kean.
Despite that, there is interest in buying the club. Former Accrington Stanley chairman Ilyas Khan has said he is prepared to front a consortium having held discussions with local investors and businessmen.
Khan told BBC Radio Lancashire: "I read the statement they [Venky's] put out last week that the family is in love with the club and want to see it through.
"If that is their intention, there is no time to be spent on this endeavour.
"If they were to decide that it's time for them to move on to a different business venture, I think our consortium is the only one that could turn this into a success for the town."