The 42-year-old, who joined Anfield in 2012, nearly guided Liverpool to the Premier League title in his second season at the club but eventually finished two points behind champions Manchester City.
Luis Suarez's departure and Daniel Sturridge's injury problems contributed to Liverpool's inconsistent form as they finished in sixth place last term, while Rodgers has now left Anfield with the team registering just three wins from their opening eight games.
Jurgen Klopp, who left Borussia Dortmund at the end of last season, is the early front runner to replace Rodgers, while Carlo Ancelotti is also being considered
