The highly-rated 22-year-old has been strongly linked with a move to Old Trafford, where he would reunite with former boss Erik ten Hag.
And, with just four days of the transfer window remaining, United – who have not strengthened in attack this summer – appear close to getting their man.
According to The Athletic (opens in new tab), the two clubs are nearing an £84m agreement for Brazil international Antony – who has been omitted from Ajax's last two matchday squads.
United won 1-0 at Southampton (opens in new tab) on Saturday to make it back-to-back Premier League victories, building on Monday's 2-1 home triumph over Liverpool.
Ten Hag still wants to improve his options as looks to turn this mini revival into something more sustained, though. After his side's success at St. Mary's, he told (opens in new tab) BT Sport:
"If a good player is available, we will strike because we want to strengthen the squad. We will be alert until the last second of the window."
But it could be a case of one in, one out in the attacking department at Old Trafford: Ten Hag's comments came as he confirmed that Cristiano Ronaldo's future remains uncertain.
"I cannot say that [Ronaldo will definitely stay at United]," the Dutchman said. "We plan with him and we stick to the plan."
Next up for United is a trip to the King Power Stadium to face Leicester (opens in new tab) on Thursday night.
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