The Red Devils were understood to have earmarked the 24-year-old as a priority signing after he won Ajax's 2021-22 Player of the Year award under Erik ten Hag.
Arsenal also showed an interest in signing the versatile defender, and he was thought to have been impressed with Mikel Arteta's project for the North London club.
The Gunners were believed to have seen two bids rejected by the Eredivisie champions for Martinez, and Arteta's side ultimately cooled their interest in a deal for the Argentina international.
As a result, Man United were able to jump to the head of the queue for Martinez's signature, and a £45m fee has reportedly now been agreed between the two clubs.
The 5ft 9in centre-back was understood to have been attracted to the possibility of working under Ten Hag again, and he has now confirmed that his arrival at the Theatre of Dreams is imminent.
On Friday night, agent and Elche chairman Christian Bragarnik posted an Instagram story in which the duo were on a private jet with the caption "heading to Manchester @lisandromartinezzz".
The former Defensa Central man reposted the story with two heart emojis as he prepares to reunite with Ten Hag, and he will soon complete his medical and finalise his visa.
Martinez would therefore leave Ajax after scoring six goals and setting up six more in 118 appearances for the Amsterdam giants, including one goal and four assists from 36 outings last season.
The South American has won two Eredivisie titles, one KNVB-Beker and one Dutch Super Cup since 2019, and he was also part of Argentina's triumphant 2021 Copa America squad.
Gunners boss Arteta was said to have been tempted by the idea of using Martinez as a left-back, while Ten Hag could also deploy the 24-year-old as a deep-lying midfielder.
Martinez should be confirmed as Man United's third summer signing imminently, following left-back Tyrell Malacia and playmaker Christian Eriksen to Old Trafford.
Eriksen's arrival on a three-year deal was announced on Friday, but a report from Metro states that neither Eriksen nor Martinez will fly out to Australia for the remainder of the club's pre-season tour.
Martinez and Eriksen will supposedly train together at Carrington before meeting the rest of the squad upon their return from Oceania, where they thumped Melbourne Victory 4-1 on Friday.
The Red Devils also secured a 4-0 win over Liverpool in Bangkok earlier this week and will face Crystal Palace and Aston Villa Down Under before heading back to the UK.
Ten Hag's side will also travel to Norway to meet Atletico Madrid on July 30 before the Dutchman takes charge of his first home game against Rayo Vallecano a day later.