The former Swansea City boss could guide the Reds to Premier League glory this season, as the club are currently four points adrift of leaders Chelsea in the table with a game in hand.
"I feel really at home here now," BBC Sport quotes Rodgers as saying. "I couldn't think of being anywhere else, it's a wonderful club.
"It's good for the supporters to dream. It's great pride for them to see their players represent the shirt so well."
Liverpool can go to one point within Chelsea if they win Saturday's match against Cardiff City.