The deal is for an initial fee of €95m (£81.3m) - which will be the fourth most expensive signing in Premier League history.
The deal includes a potential €5m (£4.27m) of add-ons - making him United’s second most expensive signing, ahead of the £80m paid for defender Harry Maguire.
Antony, 22, scored 24 goals and provided 22 assists in 82 appearances for Ajax.
He is a natural right winger and is therefore expected to compete for a starting berth with Jadon Sancho.
However, Sancho was also a fairly expensive player and has proven in the past he is a top star.
So, how will Erik ten Hag, who is clearly a big fan of Anthony as they worked together at Ajax and he has now perused United to spend big on him, fit all of Anthony Martial, Marcus Rashford, Jadon Sancho and Cristiano Ronaldo into a front-three? Well, Fabrizio Romano has provided some insight into that on Tuesday.
Fabrizio Romano said: “Now that a deal to sign Antony from Ajax is about to be officially announced, Manchester United have a really, really good attack.
“Marcus Rashford, Anthony Martial, Jadon Sancho, Antony and of course, Cristiano Ronaldo… But five does not go into three.”
“Keep an eye on Ronaldo until the end of the window,” Romano added.
“Let’s see what happens, but when it comes to Antony, Erik ten Hag views him as a regular starter but has already told Manchester United’s attackers that he will frequently rotate.”