The 18-year-old has signed a pre-contract agreement and will officially join the Reds on January 1.
Noonan has scored eight goals for Shamrock this season and featured five times in European qualifiers.
He also made history against Molde in February 2025, becoming the youngest scorer in Conference League history and second-youngest in a UEFA club competition.
"Michael is a player we, along with a number of other clubs, have followed closely for some time, so we're delighted he has chosen Nottingham Forest,” said Chief Football Officer, George Syrianos.
"Recruiting high-potential young players is central to our long-term vision for the academy.
“We want the academy to be a genuine pathway to the first team, and that means bringing in the right talent early and giving them a clear, individual plan to get there.
"We look forward to welcoming Michael and his family to Forest and to supporting his development in the years ahead."