The League Managers Association announced the move.
"The LMA can today confirm that Steve Clarke and Liverpool FC have parted company on amicable terms," the body said.
Clarke was appointed first team coach on a three-year contract in January 2011 as Kenny Dalglish's first appointment on replacing Roy Hodgson as manager.
Rodgers, the former Swansea City manager who was appointed last Friday, has yet to name a number two but three of his staff have moved with him from Wales including his assistant Colin Pascoe.
Performance analyst Chris Davies and head of sports science and medicine Glen Driscoll have also joined the Anfield club.