"We are very sorry that Sammy has decided to leave the club, but he goes with all our very best wishes for the future," said Rick Parry.
"We wanted Sammy to remain and work as part of Rafa Benitez's back room team, but totally respect the decision he has taken.
"Sammy has been a wonderful servant to Liverpool as both player and coach and he will be missed by colleagues and supporters alike.
"He should be proud of his contribution to the successes achieved at the club in recent years.
"Discussions will now be held with Rafa to identify a replacement for Sammy."