Arsenal said on their website they will be discussing other potential roles with the 56-year-old ex-Ireland international to ensure he continues his involvement with the North Londoners.
"Liam has a deep understanding of what it takes to discover and develop a talented youngster into someone who can perform at the highest level," said chief executive Ivan Gazidis.
"He has made a massive contribution to Arsenal Football Club. It will be difficult to find a worthy successor but we will be looking for someone who can build on what Liam and his team have created."
Brady started his career with Arsenal as a schoolboy in 1971 and made his professional debut in 1973.
He also played for Italian clubs Juventus, Sampdoria, Inter Milan and Ascoli before ending his career at West Ham United.
He managed Celtic and Brighton and Hove Albion before taking over Arsenal's academy where he developed current first-team players like Jack Wilshere and Kieran Gibbs as well as England defender Ashley Cole who has since moved to Chelsea.