The 20-year-old Norwegian, who missed Rovers' previous three matches, limped out of Saturday's 3-0 FA Cup fourth-round win at Derby.
"It is his hamstring again, it is the same problem," said Blackburn boss Michael Appleton. "He's done a million and one tests in terms of his strength and we thought he was right.
"He felt right, he felt great and wanted to play so it is a shame because we are going to miss his pace in this side so that is something we'll have to look at."
Former Manchester United player King initially joined Rovers on loan in November 2012 and, after scoring two goals in eight games, made the switch permanent two months later.