The 24-year-old was a key player for his Dutch club during the 2021-22 campaign, scoring once and providing four assists in 36 appearances in all competitions.
Martinez is widely expected to leave Ajax this summer, with Man United and Arsenal seemingly going head-to-head for his signature in the current market.
According to The Mirror, the 20-time English champions have now made a £39.6m offer for the Argentina international in the hope of bringing him to Old Trafford.
The report claims that the deal would see Ajax receive £35.3m up front, with a further £4.3m in guaranteed add-ons, and the defender has made it clear that he wants to move to the Premier League.
"Martinez is very high on Erik ten Hag's wishlist, so I expect big developments in the 24 or 48 hours," he told De Telegraaf. "Martinez will also have his hopes pinned on United, who are an absolute top club for him.
"Ajax will really cash out for Martinez and go to €45m (£38.7m)-€50m (£43m), but that is the advantage if two clubs bid, as we saw with Lyon and Manchester United, then it suddenly goes higher."
Martinez has played 120 times in total for Ajax since arriving from Defensa y Justicia in 2019, helping the club win back-to-back league titles and the KNVB Cup in 2021.
Ajax have seemingly managed to resist interest in Jurrien Timber this summer, but they will find it difficult to keep hold of his teammate amid the latest developments.
The 20-time English champions are said to be keen to complete a triple deal for Barcelona's Frenkie de Jong, Feyenoord's Tyrell Malacia and free agent Christian Eriksen, in addition to signing Martinez.
Man United will fly out to Thailand next Friday for their pre-season tour, with their opening match of the summer coming against bitter rivals Liverpool on July 12.