The Red Devils have been strongly linked with the 24-year-old in recent weeks, with the club thought to be going head to head with Arsenal for his signature.
Martinez was pictured flying to Manchester alongside his agents on Friday, with the 20-time English champions winning the race for his signature.
Man United have now confirmed that they have reached an agreement with Ajax to sign the Argentina international, who could report to the club's Carrington training complex next week.
"Manchester United is delighted to announce the club has reached agreement with Ajax for the transfer of Argentine international defender Lisandro Martinez, subject to medical, to player terms being finalised, and to UK visa requirements," read a statement on the club's official website.
The Red Devils will allegedly pay £46.8m plus £8.5m in add-ons for the South American, who is expected to pen a five-year contract with the club.
Martinez was a key player for Ajax last season, scoring once and providing four assists in 36 appearances in all competitions.
Dutch journalist Marcel van der Kraan recently told De Telegraaf: "Martinez is very high on Erik ten Hag's wishlist. Martinez will also have his hopes pinned on United, who are an absolute top club for him."
Martinez has played 120 times for Ajax since arriving at the club from Defensa y Justicia in 2019, helping the team win back-to-back league titles and the KNVB Cup.
At 5ft 9in, he will be one of the smallest centre-backs to play in the Premier League, and there have been claims that he could operate as a defensive midfielder for the club.
Man United are currently in Australia on their pre-season tour, but Martinez is not expected to travel to the country and is instead expected to train at the club's training complex alongside fellow new arrival Christian Eriksen.
The 20-time English champions have two more friendlies in Australia, taking on Crystal Palace and Aston Villa, before heading back to Europe to take on Atletico Madrid and Rayo Vallecano.
Man United are currently well-stocked in central defence, but Eric Bailly, Axel Tuanzebe and Phil Jones could depart before the end of the transfer window.
Tyrell Malacia became the team's first signing of the summer, and at least one more player is expected to follow Martinez to Old Trafford, with the club still chasing Barcelona's Frenkie de Jong despite his desire to remain at Camp Nou.
The Red Devils will begin their 2022-23 Premier League campaign at home to Brighton & Hove Albion on August 7, when Martinez could make his competitive debut for the club.