The Red Devils have had a series of bids rejected for the Brazilian international, with the latest rebuffed offer reported to be around €90m.
Antony joined Ajax from Sao Paulo in the 2020 winter transfer window and he has won two Eredivisie titles, both under Manchester United manager Erik Ten Hag.
Ten Hag has already raided Ajax for defender Lisandro Martinez and is also keen on a reunion with Antony.
The 24-year-old says Ajax have known for some time of his intentions to try a new challenge and wants them to accept Manchester United's offer before the transfer window closes next week.
He told transfer expert Fabrizio Romano: "Since February of this year, my agents came to Amsterdam to inform Ajax of my desire to leave the club to face a new challenge and that some interested clubs would arrive and with them certainly a great offer.
"In June of this year, I interrupted my vacation and came personally to inform the managers of Ajax, including the new coach, about my wish to leave and that they should consider this possibility, because when I started in Amsterdam it was a project for two seasons.
"During the window months, meetings continued and I also received a proposal from Ajax for contract renewal. I made it clear once again: I want to leave the club.
"Today, in a meeting with the club, I expressed my old interest in leaving the club, only this time with a considerable offer on the table. Others had already arrived. Ajax always refused and hindered by exit with the argument that they only have five days to replace me.
"I'm not asking Ajax to release me, I'm asking Ajax to sell me with the highest bid ever for a player in the Eredivisie. I've been insisting on this theme since February so that the club could rebuild the team with peace of mind.
"I was very happy in Amsterdam, I won titles at Ajax, made friends and built part of my career, but now I reinforce that I am ready and full of motivation to follow my story and my dreams.
"People need to listen to me and understand that my motivation moves me towards happiness. I need this to continue performing at a high level. Ajax will always be in my heart.
"I hope Ajax fans understand me because the future for a football player is a totally uncertain thing and the opportunities can be unique.
"My message is one of gratitude for Ajax, club and fans."
