"I would be delighted to have the job. If I do a good job, perhaps I will come into consideration," he said.
"It's an exciting time for the club. They have big plans and if I can be part of it, I will be delighted."
Blackburn became the first Indian-owned Premier League club when they were taken over by the Rao family, who own the Venky's group, in a £43m deal in November.
Although Allardyce had guided Rovers to 13th place in the table, it was not deemed good enough for the owners who sacked him after almost two years in charge at Ewood Park.
His departure led to criticism of the owners by many leading football figures amid rumours of them showing an interest appointing Diego Maradona as manager and confirmation that a fifth-place finish was an unlikely target for Rovers this term.
But Kean is hoping that his experience, which includes a spell as a coach with Reading before becoming Chris Coleman's assistant at Fulham, Real Sociedad and Coventry City and then joining Blackburn in August 2009, can help him.
"Hopefully it [the role] will be in a position that suits the bit I am good at, which is being out on the training ground, developing players and building teams. If that is the manager's job, great," he admitted.
It all depends on what happens on the pitch. If we go on a 10-match winning streak, he'll be a strong candidate to take over full-time
"If it is in another capacity, then it is something I would enjoy doing for the club."
Kean confirmed he had spoken regularly with Allardyce since the decision to dispense with his services was made.
"Sam has been very supportive. I have worked with him for the past 18 months and I have also worked really closely with Neil McDonald, who is a fantastic coach.
"You have to respect people that brought you to the club. Sam brought me here and showed a lot of confidence in me.
"He gave me my job and gave me a lot of say in what goes on at the club. I will always be grateful for that.
"It is never good when someone in our line of work loses their job," he reflected.
"I have had it on both sides. I walked in at Fulham when Jean Tigana lost his job and I had to pick up the reins.
"Then I was sacked and had to walk out and someone else came in. That is the nature of the job."
Source: BBC Sport
Source: BBC Sport