The 20-year-old picked up the injury playing in the Reds' Under-21s side against Southampton on Monday night and is now facing a lengthy spell on the sidelines.
Rodgers' side host Norwich City in the Premier League on Saturday, with Flanagan the only fresh injury concern as Jose Enrique continues to recover from a hamstring injury.
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"Even though Jon hasn't featured so much, he's been in some of our Europa League games and played in the FA Cup.
"He's out of the squad now for up to 12 weeks with a knee injury.
"Apart from that, we're pretty fine."
Liverpool sit eighth in the Premier League table and will be looking for a win against the Canaries at 15:00GMT on Saturday as they aim to bounce back from the 2-1 defeat at Manchester United.