Antony made his Brazil debut in October 2021 and has played nine times for them
It is understood United will pay an initial fee of £80.75m, with a further £4.25m in add-ons.
The 22-year-old is due to arrive in Manchester to complete a medical within the next 48 hours.
Antony would become Manchester United's joint-second most expensive signing alongside defender Harry Maguire - the £89m paid for Paul Pogba remains the club's record.
Antony scored 25 goals and provided 22 assists in 82 appearances for Ajax, having joined the Dutch side from Sao Paulo in 2020 and played under current United boss Erik ten Hag.
He made his international debut for Brazil in October 2021 and now has two goals in nine appearances.
Antony, who contributed to seven goals in his 11 Champions League appearances for Ajax, would also become the joint-fourth most expensive signing in Premier League history.
Only Pogba's move to United, Chelsea's signing of Romelu Lukaku and Manchester City's deal for Jack Grealish would be more costly.
United travel to Leicester City in their next Premier League fixture on Thursday, 1 September, but Antony would need to receive a work permit before making an appearance.
The attacker would become United's fifth signing of the summer, following the arrivals of his former Ajax team-mate, defender Lisandro Martinez, midfielder Casemiro from Real Madrid, full-back Tyrell Malacia from Feyenoord and playmaker Christian Eriksen on a free transfer.
Speculation also remains as to whether Cristiano Ronaldo will stay at the club following concerns about how the 37-year-old's mood is affecting morale in the camp.
-- Courtesy of BBC Sport