Sources told ESPN the deal is worth an initial €95 million with another €5m in add-ons, which will make it the second-most-expensive transfer in United's history behind Paul Pogba, who joined from Juventus for a then-world-record £89.3m (€105m) in 2016.
The deal is subject to a medical, player terms being finalised and international clearance.
Antony will be reunited with his former Ajax manager Erik ten Hag, who will complete his fifth signing of the summer after bringing in Tyrell Malacia, Christian Eriksen, Lisandro Martinez and Casemiro.
Antony signed for Ajax from Sao Paulo in 2020 and has made 82 appearances for the Eredivisie giants, scoring 25 goals.
He has also been capped nine times for Brazil, scoring twice.