New Manchester United boss Erik ten Hag has specifically asked the club to sign Antony, but Ajax do not want to play ball and have put a high £68m asking price on his head.
Ajax have sold a number of top stars this summer and do not want to see Antony also head out of the exit door.
The Brazilian has admirers in the Premier League beyond Manchester United though, with Liverpool and Tottenham having actively monitored him.
Both clubs may not move for Antony in the ongoing window, but if he does not join Manchester United, he would be available for them to explore in future windows.
Ajax expect Antony to stay and he will have the opportunity to further impress potential suitors should he stay put in Amsterdam.
Tottenham have been heavily active in the transfer window so far and though they are open to further signings, Fabio Paratici’s focus at present is on offloading surplus to requirements players.
Liverpool splashed the cash to land Darwin Nunez from Benfica, while Fabio Carvalho is another final third option to have been added to the mix at Anfield this summer.