The Northern Irishman moved to Leicester from Celtic in February after winning seven trophies in just under three years in charge in Scotland.
The 46-year-old has taken Leicester to second in the Premier League behind Liverpool, with 11 wins from 15 matches this season.
Leicester, the 2015/16 Premier League champions, have won their past seven league games to equal their top-flight record for consecutive victories.
Rodgers' success at the King Power Stadium saw him linked heavily with the Arsenal manager's job after the sacking last week of Unai Emery.
"When I made the decision to come to Leicester City, it was because I was excited about the direction the club was taking and how I could apply my experience to the journey," said Rodgers, who went close to winning the Premier League with Liverpool in 2014.