Martinez was Ajax's player of the year for the 2021–22 season
The 24-year-old is expected to sign a five-year contract with the option of a further year.
The move, for an initial £49m, is subject to a medical, personal terms being finalised and a UK visa.
The deal includes £8.5m in add-ons and £2.2m in solidarity payments to Martinez's former clubs.
He made 120 appearances for Ajax after moving from Argentinian side Defensa y Justicia for £6.3m in 2019, and also played for Newell's Old Boys.
Martinez has won seven caps for Argentina since making his senior debut in March 2019.
He is United's third summer signing after Tyrell Malacia joined from Feyenoord and Christian Eriksen moved on a free transfer.
The Old Trafford club said it was "delighted" to announce an agreement had been reached with Ajax.
"The transfer is still subject to a medical examination and the contractual agreement of both clubs with the player," said the Dutch club in a statement.
"The defender is still under contract with Ajax until 30 June, 2025. Ajax agreed a transfer fee of 57.37m euros with the English side. That amount can increase to 67.37m euros through variables."
Martinez played under new United manager Erik ten Hag at Ajax and won the Eredivisie title with the club in 2021 and 2022, as well as the Dutch Cup in 2021.
He was also Ajax's player of the year for the 2021-22 season.
-- Courtesy of BBC Sport