The Argentine centre-back, 24, is a favourite of new Red Devils boss Erik ten Hag following their time together at the Eredivisie giants.
It is understood Ten Hag wants a deal wrapped up as soon as possible, with the Gunners also trying to secure the highly-rated defender.
United hope the Ajax connection between Martinez and Ten Hag will give them an edge over Arsenal.
The Red Devils have already signed left-back Tyrell Malacia from Feyenoord, with the player joining for £13million.
And a contract has also been agreed for free agent Christian Eriksen who is set to arrive on a three-year deal.
Both of those incomings were strongly recommended by Ten Hag as he looks to make his mark immediately.
Barcelona and Netherlands midfielder Frenkie de Jong also remains linked with a £56m move to Old Trafford.
Elsewhere, the Red Devils face a battle to keep star forward Cristiano Ronaldo.
The Portugal sensation, 37, has told the club he wants to leave this summer, with Chelsea among clubs thought to be considering a bid.
talkSPORT understands Ronaldo remains a big part of Ten Hag's plans next season and United have no intention of selling the goalscoring legend.