“I am, of course, incredibly disappointed to be leaving Liverpool Football Club. It has been both an honour and a privilege to manage one of the game’s great clubs for the last three years," Rodgers, 42, said.
“I have worked every day to represent the club to the best of my ability, to develop both individual players and a team that the club’s magnificent fans can be proud of.
Rodgers, who almost led Liverpool to the title in 2014 when they were runners-up, is their first manager, besides current England boss Roy Hodgson, to fail to win a trophy in the first three years of his reign since Phil Taylor in the mid-1950s.
"There have been some very memorable moments during my time at Liverpool and I would like to thank all of the players for their hard work and commitment," Rodgers added in a statement released by the League Managers Association.
"The current squad is one in transition, but they have some real talent and are showing a strong sense of togetherness.
"I expect to see them continue to grow and develop over the coming weeks and I wish them and my successor well for the rest of the season."
Liverpool have been linked with ex-Borussia Dortmund coach Juergen Klopp and former Real Madrid boss Carlo Ancelotti.
Former Swansea city manager Rodgers, who succeeded Liverpool icon Kenny Dalglish at Anfield in June 2012, leaves the club in 10th place with three wins from their first eight league games.
(Writing by Ken Ferris, editing by Pritha Sarkar)
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