Shaun Wright-Phillips insists he is happy again at QPR after turning down "offers and opportunities" to move in January.
Blackburn goalkeeper Paul Robinson has told Sky Sports he will leave the club this summer and could move abroad.
The 56-year-old former Republic of Ireland midfielder made more than 200 appearances for the Gunners between 1973 and 1980, winning the FA Cup in 1979, before joining Juventus where he went on to claim the Serie A championship twice.
Brady - who also had spells in management with Celtic and Brighton, as well as being assistant to Ireland boss Giovanni Trapattoni - took over as Arsenal's head of youth development and academy director in July 1996, helping bring through the likes of current England internationals Jack Wilshere and Kieran Gibbs.
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."