The former international will quit his post as head of youth academy at the Premier League club in May 2014, a job he has held since 1996, bringing through a host of first-team regulars.
The Dubliner played a key role in bringing the Italian into the Ireland job and served as his assistant for the first two years of his reign, leaving the job after the infamous play-off defeat to France in Paris in 2010.
Trapattoni's current deal with Ireland runs until the 2014 World Cup in Brazil and the 73-year-old is not expected to extend his six-year stay with the FAI. The Italian is under pressure to deliver results after a poor European Championships last summer and a difficult start to qualifying.
The timing could be perfect for the 56-year-old Brady (pictured), who has worked with many of the players and understands the set-up.
Mick McCarthy remains the bookies' favourite, but Brady is believed to be in the mix.
The next manager will also be responsible for ushering in a new era as Robbie Keane and Richard Dunne are expected to follow Shay Given and Damien Duff into retirement and the words of Arsenal chief executive Ivan Gazidis should serve as a ringing endorsement if he chooses to try and take on the job of managing a country he represented 72 times.
"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 FC," the chief executive said.
"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."