The Old Trafford giants are understood to have seen a bid of around 90million euros (£76m) rejected for the Brazilian forward last week but have now increased their bid for the 22-year-old.
The new proposed transfer fee for the Brazil international is reportedly 100m euros (£84.9m) including add-ons - news that will please Antony, who made his frustrations clear in an interview on Friday.
"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," Antony said via Italian journalist Fabrizio Romano on YouTube.
"During the window months, meetings concluded including a proposal from Ajax for a contract renewal. I made it clear once again I want to leave the club.
"Today in a meeting with the board I expressed my interest in leaving the club, this time with a considerable offer on the table. Others have already arrived (but) Ajax always refuses with the argument it only has five days to replace (me).
"I'm not asking (for Ajax) to release me, I am asking Ajax to sell me for the highest bid received for a player who plays in the Eredivisie."
Antony added: "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."
Anthony was left out of Ajax's match day squad for the 2-0 win over Utrecht on Saturday.
United paid an initial 57.37million euros (£48.3m) for the Argentina international - a fee that could rise by a further 10m euros to around £56.7m.