Antony was in impressive form for Ajax last term, scoring 12 goals and registering 10 assists in 33 appearances in all competitions, including eight goals and four assists in their title-winning Eredivisie campaign.
Man United boss Erik ten Hag is believed to be desperate for a reunion with the Brazil international, who has represented his Dutch side on 78 occasions since arriving from Sao Paulo in July 2020, contributing 22 goals and 20 assists.
According to Goal, the 20-time English champions have offered €60m (£51m) for the forward, but Ajax are holding out for a fee of €80m (£68m) due to his importance to the club.
The report claims that Antony's agents are putting pressure on the Dutch champions to accept a lower fee, as the player is desperate to move to Old Trafford before the end of the transfer window.
"It is clear that we want to keep that boy. I think he still has a contract until 2025. I really want to work with him; he knows that, and the club knows that too," new Ajax boss Alfred Schreuder recently told ESPN.
A move for the attacker at this point would also be complicated due to an ankle problem, which has kept him on the sidelines since March.
"He is still recovering from his injury. But I think the next few weeks are important. For his injury, we hope it improves quickly. I assume that he will be fit at the end of the preparation," Schreuder said when asked for an update on his fitness.
Bayern Munich are among a number of other clubs to be credited with an interest in the South American since the end of last season, but he is believed to only be interested in a reunion with Ten Hag in Manchester.
"I want to thank you for everything you taught me. Know that I will always stand with you if you need me," Antony posted on social media when Ten Hag left to take the Red Devils job.
Man United are also believed to be in talks with Ajax over Lisandro Martinez, with a recent report claiming that a deal for the Argentina international was close to completion.
Tyrell Malacia became the club's first signing of the summer on Tuesday, while Christian Eriksen is expected to arrive on a free transfer, potentially before the end of the week.
A move for Barcelona's Frenkie de Jong has stalled, though, and Man United are also having to deal with the speculation surrounding Cristiano Ronaldo, who has allegedly asked to leave Old Trafford.
Ten Hag is believed to view Antony as the ideal replacement for Ronaldo, with the latter reportedly instructing his agent Jorge Mendes to attempt to negotiate a move away from the Red Devils.
Man United will head out to Thailand on Friday for the first leg of their pre-season tour, and at this point, it is unclear whether Ronaldo will be a part of the squad that travels to Bangkok.