The deal is worth an initial €95 million with another €5m in add-ons, making the winger United's second-most-expensive signing ever once the transfer is completed. The deal is subject to a medical and the UK's visa process.
Antony is set to sign a five-year contract with the option of another year.
Ten Hag has been keen to bolster his forward line this summer, particularly after Anthony Martial picked up an achilles injury.
The France international missed the first two games of the season with a hamstring problem sustained during preseason.
The winger, capped nine times by Brazil, scored 12 goals in 33 games for Ajax last season.
He arrived in Amsterdam from Sao Paulo in 2020 for an initial €15m fee.
The 22-year-old was left out of the Ajax squad for their 1-0 win over Sparta Rotterdam last weekend and also missed the fixture against Utrecht on Sunday, while United remained locked in talks with the Dutch champions.