The Gunners have already made a number of additions to their first-team squad this campaign, including securing a deal for Porto playmaker Fabio Vieira.
Earlier this month, it was claimed that the North London outfit failed with their opening bid for Martinez, who can play at centre-back and in defensive midfield.
Nevertheless, according to the Daily Mail, Arsenal have lodged a much higher offer than their first proposal which was said to be worth £25m.
The Premier League side are said to have offered Ajax a fee of £34m, a proposal which is still allegedly lower than Ajax's valuation.
The Dutch giants are allegedly hoping to recoup as much as £43m for the Argentina international, who made 24 appearances in the Eredivisie last season.
A potential stumbling block is Ajax's lack of necessity when it comes to selling another player with key man Ryan Gravenberch having already moved to Bayern Munich.
However, it is suggested that any formal bid will be put on hold until United are able to sell or part ways with at least one of their current defenders.
A recent report has claimed that United's transfer budget will be restricted to £100m unless player sales are made over the coming weeks and months. body check tags ::