The 24-year-old was in strong form for his Dutch club during the 2021-22 campaign, scoring once and providing four assists in 36 appearances in all competitions.
Martinez's future has been the subject of much speculation in recent weeks, with both Arsenal and Man United said to be keen to bring him to the Premier League for the 2022-23 season.
A recent report claimed that Man United were preparing a £40m offer for the Argentina international, identifying him as their leading centre-back target, having missed out on a deal for his teammate Jurrien Timber.
However, according to Italian journalist Fabrizio Romano, Arsenal's interest is "more advanced", with Erik ten Hag's side currently focusing on securing deals for Barcelona's Frenkie de Jong and Feyenoord's Tyrell Malacia.
"There's also interesting news emerging on the future of Ajax defender Lisandro Martinez. The impressive Argentine has been linked by various sources with both Arsenal and Manchester United, with slightly differing takes on what's going on with the player," Romano wrote in his column for CaughtOffside.
"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.
"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."
Martinez still has more than three years left to run on his contract in Amsterdam, and it was recently claimed that Ajax value him in the region of £50m.
"He (Martinez) is an expensive one because Ajax are demanding £45m-£50m for this player. Ajax have no need to sell anymore players," Dutch journalist Marcel van der Kraan told Sky Sports News.
Martinez has won two Eredivisie titles and one Dutch Cup during his time at his current club, making 118 appearances since arriving from Defensa y Justica in 2019.
Man United are believed to be attracted by the South American's versatility, as he is capable of playing as a left-back and in central midfield.
However, the 20-time English champions could be beaten to his signature by Arsenal, with the Gunners looking to follow Gabriel Jesus's signing with the capture of the Ajax centre-back.
Man United have also been credited with an interest in Villarreal's Pau Torres, although talk of a possible move for the Spain international has quietened over the last couple of weeks.