Antony, 22, was identified by United manager Erik ten Hag as his top target in the attack at the beginning of the transfer season, and Ajax is holding out for a bid of approximately €100 million (£84.15 million).
If United hadn't already given Ajax £46.8 million up front for center-back Lisandro Martinez, Antony could have been acquired for €80 million.
Ajax is in a good negotiating position going into the deadline on Thursday because they also received £16 million from Bayern Munich for Ryan Gravenberch and £29 million from the transfer of Sebastian Haller to Borussia Dortmund.
United have so far failed to strengthen up front, with playmaker Christian Eriksen their only attack-minded addition so far this summer.
Ten Hag wants to recruit a left-footed forward as there is an acceptance at United they have too many attackers of an identical profile. Jadon Sancho, Marcus Rashford, Anthony Martial and Anthony Elanga have all enjoyed their finest runs of form from the left wing.
Antony has scored 21 goals for Ajax since arriving from Sao Paulo two years ago but was omitted from the matchday squad for their weekend win at Sparta Rotterdam amid his desire to join United.
“I want to keep everyone and I want Antony to stay,” Ajax coach Alfred Schreuder told ESPN on Sunday. “I don’t expect him to be sold.”
Antony was captured watching United’s 2-1 victory over Liverpool on Monday night in an Instagram story with his agent, Emerson Santos.
United had a bid rejected for Bologna forward Marko Arnautovic earlier this month before they withdrew their interest following a backlash from supporters on sporting and integrity grounds.
Ten Hag has primarily favoured targeting players he is familiar with, having approached Ajax centre-back Jurrien Timber before he opted to stay with the Dutch champions.
United have failed to make any headway in their ordeal to sign Frenkie de Jong from Barcelona, despite agreeing a fee of €85m last month.
De Jong’s wage deferral halted United’s progress on a deal and the 25-year-old, who started in Barcelona’s 4-1 win over Real Sociedad on Sunday, would prefer to stay at the club.