In a bid to close the gap to the teams which finished in the top four of the Premier League table, Mikel Arteta has been proactive with his transfer business this summer.
As well as already securing the £34m signing of Porto midfielder Fabio Vieira, it is expected that Gabriel Jesus will depart Manchester City for the Gunners.
However, despite being able to call upon William Saliba after his loan spell with Marseille, Arteta is keen to bring in another defensive-minded player over the coming weeks.
Speculation persists that the Spaniard wants to wrap up a deal for Martinez, who may be allowed to depart Ajax for the right price.
However, according to the Daily Mail, Arsenal have failed with their second approach for the versatile Argentina international.
The report claims that the North London outfit forwarded a proposal worth in the region of £38m, with a number of add-ons to follow in future years.
Nevertheless, it is claimed that Ajax opted to shun the improved offer, instead looking to hold out for an alleged asking price of £43m.
Manchester United are also said to be monitoring the situation with Erik ten Hag having worked with the player during his time at the Eredivisie champions.
Despite his ability to deputise at left-back or in a defensive midfield role, Martinez was purely used in the middle of Ten Hag's backline in the Dutch top flight last season.
The 24-year-old made as many as 23 starts, but he missed the majority of the final two months of the season due to injury.
While Martinez is believed to be open to a transfer to Arsenal, a previous report has claimed that he would prefer to make the switch to Old Trafford.
However, United are seemingly prioritising other deals at this stage with Tyrell Malacia, Frenkie de Jong and Christian Eriksen also said to be on their radar.
Martinez still has three years remaining on his contract having penned fresh terms as recently as October.