Liverpool hope to appoint Jurgen Klopp to replace by the end of the week, the reports.
The former Borussia Dortmund coach is favourite to take over from the Northern Irishman, with ex-Chelsea boss Carlo Ancelotti also a contender.
German Klopp, 48, is understood to be open to the idea of a move.
Talks with his representatives are progressing well and the club want a new manager in place to prepare for a visit to Tottenham on 17 October.
In a statement issued by the League Managers’ Association on his behalf, Rodgers, 42, said he was “incredibly disappointed” his three-and-a-half-year tenure at Anfield was over.
Rodgers, who said he is in no rush to take another job, added the Reds squad was in transition but was showing a “strong sense of togetherness”.
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