The reasons for the Anfield legend's departure are unclear and the club have yet to comment on the reports , but the appointment of Steve Clarke last season may have minimalised his role.
Lee, who had been working as part of Kenny Dalglish's staff, has already been linked with a move to West Ham, where he would be reunited with former Bolton colleague Sam Allardyce.
The ex-Liverpool player had been a part of the coaching staff since being appointed as reserve team coach by Graeme Souness in 1993.
He was promoted to a first-team role under Roy Evans and kept his position when Gerard Houllier became the sole manager.
Lee left Liverpool in 2004, when he took up a full-time position with England before becoming Allardyce's assistant at Bolton.
The now-52-year-old succeeded Allardyce as manager at Bolton, but was sacked after winning just one match in 11 and returned to Anfield in 2008 as a deputy to Rafa Benitez.