The former Republic of Ireland international, who has been part of the Gunners' set-up for the last 17 years, has established the academy as one of the best in the world and developed numerous professional players including Jack Wilshere and Kieran Gibbs.
Brady, who starred for the Gunners in the 1970s before moving to Italy where he enjoyed spells with Juventus, Sampdoria, Inter Milan and Ascoli before ending his career with West Ham, has given an early indication of his desire to leave the role in order to help recruit a successor.
After a glittering playing career, Brady also had spells as manager of Celtic and Brighton, as well as assistant manager of the Republic of Ireland.
Arsenal chief executive Ivan Gazidis said: "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.
"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."