Manchester United have been strongly linked with the 22-year-old this summer, with Red Devils boss Erik ten Hag a huge admirer of the player that he managed in Amsterdam.
However, the 20-time English champions have recently been focused on securing deals for Christian Eriksen and Lisandro Martinez, while they continue to chase Barcelona midfielder Frenkie de Jong.
Ajax's asking price is also thought to be a huge issue for Man United this summer, with the Dutch champions said to value the Brazil international in the region of €80m (£67.8m).
According to Football Insider, Liverpool have made contact with Ajax to discuss the possibility of signing the right-sided forward during this summer's transfer window.
The report, though, claims that the Reds are also not planning to meet the Dutch side's asking price for the South American, having already spent big on Uruguay international Darwin Nunez this summer.
Chelsea have also been credited with an interest in Antony, who still has another three years left to run on his current deal in Amsterdam.
The forward was in strong form at club level 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.
Meanwhile, he has represented his Dutch team on 78 occasions since arriving from Brazilian outfit Sao Paulo in July 2020, contributing 22 goals and 20 assists in the process.
Ajax boss Alfred Schreuder recently insisted that his side are determined to keep hold of Antony this summer amid the speculation surrounding his future.
"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," Schreuder recently told ESPN.
An ankle problem has kept Antony on the sidelines since March, meanwhile, with the forward not yet involved for the Dutch side in pre-season.
"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.
Ten Hag's close relationship with Antony is thought to place Man United in a strong position when it comes to his signature, but the Red Devils could struggle to afford him in the current market.
"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 charge at Old Trafford.
Ajax have already made three big sales this summer, with Ryan Gravenberch, Sebastien Haller and Lisandro Martinez moving on.