Page was dismissed by the Football Association of Wales in June last year after the Europeans missed out on Euro 2024 qualification and drew with minnows Gibraltar.
The 50-year-old replaces Barry Lewtas, the Reds' longtime youth coach who left the club this summer after four years.
"Rob brings a wealth of playing and coaching experience to a key role as young players look to bridge that gap from Academy prospect to first-team footballer,” said Liverpool academy director Alex Inglethorpe.
"His recent achievements in guiding Wales to the last 16 at Euro 2020 before qualifying for the 2022 World Cup speak for themselves but perhaps less heralded is the work that took place during his time in charge of the national team, which saw numerous pathway players make it into the senior squad.
"This is something we have seen first-hand at Liverpool with several of our Academy players making significant progress on the international stage under Rob’s guidance.
"This level of experience in transitioning young footballers towards the professional game and on to the highest levels of domestic and international football will be a significant resource for all at the Academy, especially the players.
"Myself and all the staff are greatly looking forward to working with Rob as we endeavour to continue producing players capable of playing at the highest level and in the most demanding of arenas."