The 21-year-old, who has progressed through the youth system at Ewood Park, is attracting attention from a number of Premier League sides with a string of quality performances this season.
Kean has since acknowledged the Lancashire outfit may be forced to sell the youngster in January, with the club keen to avoid losing him on a free in the summer.
“Any player does when it gets down to six months,” he said to reporters on Friday.
“What we have to do is to continue to be open and honest with Junior and say 'we want you to stay'.
“If he would like to commit his future to the club we have always said from day one that we would like to get it tied up.
“He has probably taken the opportunity to look elsewhere. I don't know where he is on it. If we can get to a point where he wants to sign then great.
“If not, if there is a bid that comes in then we would obviously have to consider that, and the fact that he hadn't signed. But no bids have come in.”
The 44-year-old boss also rued the decision to only give the young winger a two-year deal back in 2010 when Sam Allardyce was in charge at Ewood Park.
“A couple of years ago when we only signed him for two years we should have made that a longer contract,” the manager continued.
“Everybody would say we should have pushed to make it longer because now when there's six months left it's tough for the club.”
Blackburn face Newcastle United in the third-round of the FA Cup on Saturday and Kean believes Alan Pardew’s side will not be too greatly affected by the loss of Demba Ba and Cheikh Tiote to the African Cup of Nations.
He said: “They have been big players for them but look at their squad and they have a number of different options.
“Yes you miss your big players but they have people in there who are still able to deliver some quality.”