Antony has 31 goals and 27 assists in 134 club appearances for Ajax and Sao Paulo, and two goals and two assists in nine senior appearances for his native Brazil.
He has won two Eredivisie titles and a gold medal in the 2020 Summer Olympics.
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 finalized 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 9 times for Brazil, scoring twice.