Antony, 22, was a standout for Ajax under Ten Hag, making 79 games and adding 22 goals and 20 assists in the process.
United executives have traveled to Amsterdam, according to The Sun, to work out a deal for the forward, who may be Ten Hag's first summer signing.
The reported compensation for bringing Antony to Old Trafford is £40 million, with the winger previously indicating that he would like to reunite with his old Ajax boss.
Antony is cited as saying in a goodbye statement sent by players and colleagues who have worked with Ten Hag:
After a successful season at Ajax last season, the Brazilian has established himself as one of Europe's most admired wingers.
He made 32 appearances for the Eredivisie champions, scoring 12 goals and assisting 10 times while contributing significantly to the team's title and KNVB Cup triumphs.
Erik ten Hag has been associated to a number of players he has previously coached, including Jurrien Timber and Lisandro Martinez, Antony's teammates.
Timber, 21, is said to have surprised Manchester United manager Jose Mourinho by turning down the option to join him at Old Trafford this summer (per Metro).
Meanwhile, according to the Daily Mail, United will compete with Arsenal for the signing of Martinez, 24, as the Red Devils try to bolster their defense.
Antony set to become the first signing of Manchester United's rebuild
Following a poor season, Manchester United is in desperate need of reinforcements this summer.
The Red Devils finished sixth in the Premier League and were knocked out of all cup competitions early.
The 13-time Premier League champions finished with the lowest point total in the Premier League era.
Erik ten Hag has been tasked with reviving a shattered United team, and Antony appears to be the first player to arrive in the new era.
Throughout last season, Manchester United had issues on the right side of their attack.
Jadon Sancho, 22, joined from Borussia Dortmund last summer but struggled on the right wing and was eventually moved to the left.
Anthony Elanga, 20, had a breakout season at Old Trafford but expressed a desire to play on the left wing while on international duty.
Marcus Rashford, 24, has never been comfortable playing on the right side of Manchester United's offense.
Mason Greenwood, 20, has been suspended by the club until the outcome of his sexual assault trial.
Ten Hag now has significant alternatives coming into next season, when he will need all of his forwards to be blazing on all cylinders.
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