Allardyce had also been linked with the managerial vacancy at Sunderland.
He was highly successful during eight years at the helm of Rovers' Lancashire rivals Bolton, establishing them in the Premier League during his time in charge at the Reebok.
The Trotters also reached the final of the Carling Cup in 2004 and qualified for the Uefa Cup for the first time in their history in 2005.
Rovers play successive Premier League fixtures against Newcastle and Bolton in January.
Former Blackburn striker Kevin Gallacher told BBC Sport: "It is not a surprise choice.
"Blackburn needs someone who can put the club back together again and I think he can do that."
Allardyce has also managed Blackpool and Notts County.
Allardyce can resurrect Blackburn - Gallacher
At Bolton he was renowned for innovative thinking, using scientific techniques to keep players fit.
Veteran Sheffield United midfielder Gary Speed, who played for Allardyce at Bolton, said: "He gets the best out of players, he's a very approachable, likeable man and I'm sure the players will enjoy his training and his company.
Regarding the style of football Allardyce would favour at Blackburn, Speed said: "He's clever enough to know if you haven't got the players to play direct, you don't play direct."
Graeme Souness, who managed the club from 2000-2004, had also been linked with the job but had already ruled himself out of the running.
"I'm afraid you are talking to the wrong man," Souness told the Lancashire Evening Telegraph newspaper.
"I don't know anything and have had no contact at all from Blackburn Rovers. I'm not your man, I can definitely say that."
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