The England forward, who suffered the injury during training on Friday, has scored 11 goals in all competitions for Liverpool this season and has linked well up front with Luis Suarez in the Anfield attack.
Rodgers told Sky Sports before his team's match at Hull: "Daniel is probably going to be out six to eight weeks. It was something we thought was never going to be as serious as that to be honest.
"He slipped over in training and we thought initially it was his toe. He got brought inside straight away and he was taken for a scan straight away and it showed he'd actually got a ligament strain. It's a big blow for us because he's been brilliant since he's come in."