The Red Devils have already had a £42million bid for the Argentine international turned down, just off the Eredivisie champions' asking price of £46m.
Now, though, United chiefs are due to have further talks with Ajax in Amsterdam on Wednesday and believe they are close to a breakthrough.
United are prepared to pay £46m, inclusive of add-ons, for Erik ten Hag's top defensive target.
Arsenal had two bids rejected by Ajax earlier this summer but are now resigned to missing out on the 24-year-old.
Ajax were originally unwilling to entertain offers for their defender after losing Ryan Gravenberch and Noussair Mazraoui to Bayern Munich and Sebastien Haller to Borussia Dortmund.
But they've now changed their stance, with Shakhtar Donetsk's Ukraine international Mykola Matvienko being eyed up as a replacement at centre-back.
Of course, Red Devils boss Ten Hag is well acquainted with Martinez following his time as his manager at Ajax before he left for United this summer.
Martinez made 36 starts in all competitions under the Dutchman last season, with any games missed mostly being down to injury issues, and played a key role in the club's Eredivisie title win.
The defender originally joined Ajax in 2019 with Ten Hag envisioning him as a midfielder, describing Martinez as the 'brain' he'd been looking for.
Although he settled in the centre-back position, Martinez's versatility is a boost for any manager he plays for, with the Argentine capable of playing at full-back and defensive midfield.
And should the talks with Ajax go well on Wednesday, a deal will be the icing on the cake of a good week for United following their 4-0 pre-season win over Liverpool.
Jadon Sancho, Fred, Anthony Martial and academy star Facundo Pellistri got the goals as the Red Devils put their old rivals to the sword, despite a whole host of substitutions throughout the game for both teams.
Those changes to the team, with 22 United players fielded across the 90 minutes, saw Ten Hag experiment with his options, and also start the game with what possibly would be his first-choice XI this season.
But the manager has stressed that United still need strengthening in both midfield and attack.
"We want to have the right players," Ten Hag said this week.
"We have a really good squad, we are working on to develop our way of playing and if the opportunity is there we will strike [to sign new players].
"I said already definitely we are searching for players in midfield and I think that we have a need in offence. We are still working for players."