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”.