Both players starred under Erik ten Hag at Ajax last term, and it is understood that the new Red Devils manager is keen to bring the duo to Old Trafford ahead of the 2022-23 campaign.
Man United have allegedly offered €50m (£42m) for Martinez, having been informed by the player's representatives that the fee would be enough to convince Ajax to sell.
However, according to 90min, the Dutch champions have now raised their asking price to €60m (£50.8m), which has angered the Argentina international.
The report claims that Arsenal have backed away from the deal due to the cost involved, but Man United are still keen to sign him despite Ajax changing their stance due to the interest in his services.
Martinez has played 120 times for Ajax since arriving in Amsterdam from Defensa y Justicia in 2019, helping the club win back-to-back league titles and the KNVB Cup.
The South American was a vital player for his Dutch club last season, meanwhile, scoring once and providing four assists in 36 appearances in all competitions.
Martinez has allegedly made it clear that he wants to move to the Premier League this summer, but Ajax are in a strong position when it comes to his future, as they have him on a deal until June 2025.
Meanwhile, the same report claims that Antony's asking price is €80m (£67.8m), meaning that Man United would have to pay €140m (£118.6m) to sign both players this summer.
Antony 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.
"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 stage of the window would also be complicated due to an ankle problem, which has kept him on the treatment table 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.
Antony has represented his Dutch side on 78 occasions since arriving from Sao Paulo in July 2020, contributing 22 goals and 20 assists, and he was also linked with Bayern Munich earlier this summer.
Man United have only managed to sign one player thus far this summer despite their struggles last season, with left-back Tyrell Malacia making the move to the club from Feyenoord.