King, 20, suffered a reoccurrence of the injury after just eight minutes in last Saturday's 3-0 FA Cup win over Derby County.
The former Manchester United forward has been struggling with the problem for the last month and scans have now revealed an extended spell on the sidelines is required.
"Josh looks a bad one to be fair," Rovers boss Michael Appleton told the club's website.
"It is a grade two hamstring injury which isn't great so that could possibly be a 6-8 week job.
"Dickson Etuhu is probably about ten days away from training whilst Dunny has had a knock in training so he has missed a few days.
"Adam Henley is probably a month away because it was also a bad hamstring injury that he did."