United have been trying to sign the Brazilian, who played under their current manager Erik ten Hag at Ajax.
Antony registered 12 goals and 10 assists in 33 games in all competitions for the Dutch side last season as they won their third consecutive Eredivisie title and reached the round of 16 of the Champions League.
"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 was quoted as saying by Italian journalist Fabrizio Romano.
"During the window months, meetings concluded including a proposal from Ajax for contract renewal. I made it clear once again I want to leave the club," Anthony said in remarks Romano read on YouTube.
"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).
The transfer window closes on Thursday.
"I'm not asking (for Ajax) to release me, I am asking Ajax to sell me as the highest bid received by a player who plays in the Eredivisie," Antony added.
United are 14th in the Premier League after three games and play away to Southampton later on Saturday.
