The Brazil international has been sidelined for the Reds after suffering a thigh injury in the Merseyside club's Premier League match against Manchester City in August.
"Lucas has been out on the field in the last week or so which is great," Rodgers told the Liverpool Daily Post. "We're starting to open up his running.
"He's not doing multi-directional movements yet but he's doing a lot of straight line running and endurance work. I'm anticipating maybe another three weeks but his professionalism is great and he'll get back probably sooner than is anticipated."
Lucas was previously out of action for the Reds for seven months after suffering cruciate ligament damage in his knee last season.