The 25-year-old suffered the problem in training after playing in the Three Lions' 1-0 victory over Norway at Wembley in a friendly last Wednesday.
The injury blow has led Rodgers to question the conduct of England's medical staff because he believes that the problem could have been prevented.
"We're disappointed because we feel it's an injury which could have been prevented," Rodgers told talkSPORT. "Clubs work differently to international teams, it's the recovery strategy. When we look at our players here, we look at them individually, in terms of what their needs are.
"Some players are very fast, they will have a second day, active recovery, whereas other players can work on that [second] day. When you're a type of player like Raheem Sterling or Daniel Sturridge, you need to recover them differently."
It was suggested that Sturridge could be sidelined for up to four months, but the latest reports have claimed that a three-week layoff is likely.