Fallows has been with the Reds since 2012, having arrived after spells with Manchester City, Newcastle United and Bolton Wanderers.
He was instrumental in helping build Jurgen Klopp's Premier League and Champions League winning teams.
Fallows intended to leave at the end of last season, but chose to stay as Arne Slot's new management team arrived to help the transition.
"I have been very privileged to work for this incredible football club,” said Fallows to the club's website. “The fantastic people here starting with the supporters make it so special – to have had the opportunity to work with so many talented players and colleagues past and present who are now friends is something I am very grateful for.
“The leadership here, from ownership, through to Michael (Edwards), Richard (Hughes), Arne (Slot), Billy (Hogan) and Alex (Inglethorpe), is world-class. I have no doubt they will continue to thrive. They are supported by great people across many departments who I will miss working with but will continue to stay close to.
“There are far too many to mention but I am especially grateful to Barry Hunter and all those within the recruitment department who have been a joy to work alongside. Leaving in the knowledge that I will be leaving this place in such great shape is something that means a lot to me.
“I look forward to supporting the team from a different perspective now but with the same passion for their success.”
Liverpool’s sporting director, Richard Hughes, said: “David’s reputation is second to none in his field and his achievements at the club mean he leaves with utmost gratitude and affection.
“It speaks volumes about Dave that he remained with us to help with the transition and settling-in of a new football leadership team, including myself. I will always be grateful for his advice and assistance during this period.”