Premier League rivals Arsenal are also interested in the versatile Argentine, but it is believed the 24-year-old is keen to reunited with his former manager Erik ten Hag at Old Trafford.
Arsenal have already made two offers for Martinez and are awaiting a response from the Dutch club after tabling a second bid worth £35m.
The defender has made more than 100 appearances for Eredivise champions Ajax, showing his versatility by playing in four different positions since joining them from Defensa y Justicia in 2019.
He predominantly played as a centre-back last season despite being just 5ft 9in, but can also play at left-back, right-back, and as a defensive midfielder.
New United boss Ten Hag worked with him during his time in the Netherlands and is one of the many Ajax stars linked with a move to join him in Manchester.
The pair have won back-to-back titles together and the Dutch boss is said to be a big admire.
Ten Hag highlighted his ability to play in various positions when there were reports that left-back Nicolas Tagliafico nearly left Ajax two years ago.
The new United boss said: "In preparation, I will try Lisandro Martínez in that position. He has also played in midfield for us, but he is a defender."
"I find Lisandro Martinez an excellent player, he is actually starting material and this is why I talk with him a lot," he added.
"But he has a lot of competition with Daley Blind, Tagliafico and [Ryan] Gravenberch. I can only start 11 players."