The Argentina international's future has been the subject of much speculation in recent days, with Arsenal and Man United both believed to be keen to bring him to the Premier League this summer.
The 24-year-old scored once and provided four assists in 36 appearances in all competitions for Ajax last term, playing a key role in their Eredivisie success under Erik ten Hag.
There have been contrasting reports when it comes to the battle for the centre-back, with Italian journalist Fabrizio Romano claiming that Arsenal are leading the race for his signature.
"My understanding is that these are two negotiations at different stages: Arsenal have already submitted official offers up to €40m (£34.2m) plus add-ons for Martinez, while Manchester United are currently busy with Tyrell Malacia and Frenkie de Jong, but have had contact with Lisandro's agents," Romano wrote in his column for Caught Offside.
"The player is so far open to negotiate with both clubs. Arsenal are insisting a lot this week because Arteta is understood to be a big fan of Lisandro, considered a versatile player with top team skills. For now, the Gunners' interest is more advanced, but this still looks like it could be an open race."
However, Dutch journalist Marcel van der Kraan has said that Man United are in pole position, with the 20-time English champions preparing to accelerate their interest in the South American.
"Martinez is very high on Erik ten Hag's wishlist, so I expect big developments in the 24 or 48 hours," he told De Telegraaf. "Martinez will also have his hopes pinned on United, who are an absolute top club for him.
"Ajax will really cash out for Martinez and go to €45m (£38.7m)-€50m (£43m), but that is the advantage if two clubs bid, as we saw with Lyon and Manchester United, then it suddenly goes higher."
According to Goal, Man United have held positive talks over a deal for the defender, with Ten Hag desperate for a reunion with the player at Old Trafford.
Arsenal are believed to have already made two bids for Martinez, with the latest one being worth in the region of £35m plus add-ons, but Ten Hag's presence at Man United could give the Red Devils the edge.
The centre-back has featured on 120 times in total for Ajax since arriving from Defensa y Justicia in 2019, helping the club win back-to-back league titles and the KNVB Cup in 2021.
The South American still has more than three years left to run on his contract, but Ajax could be open to selling him this summer, having allegedly managed to convince Jurrien Timber to remain at the club.
Harry Maguire, Raphael Varane and Victor Lindelof are all expected to remain at Man United in the current market, but Eric Bailly, Phil Jones and Axel Tuanzebe could depart, which would open up space for another centre-back to join the club ahead of the new season.