The Gunners have made two bids for the Argentina international and have now put in a third bid worth around £35 million, rising to £40 million with add-ons, according to The Telegraph.
While Mikel Arteta has identified him as a top target, Arsenal face competition from United, and Erik ten Hag is eager to sign one of his former Ajax stars.
Ajax, meanwhile, is in no rush to sell Martinez with a bidding war between Arsenal and United likely to benefit the Dutch giants.
The report adds that Arsenal hope to close a deal while United’s focus is elsewhere.
The Manchester club remain in talks with Barcelona over a transfer for Frenkie de Jong, while they are also finalizing a deal to sign Feyenoord full-back Tyrell Malacia.
Ten Hag is still keen to attract a defender and versatile attacker this summer with another Ajax star in Antony who is also on his wanted list.
Arsenal have already made three additions to their roster this summer and are expected to confirm the signing of Gabriel Jesus this week.
The Gunners had hoped to sign Raphinha, the star of Leeds United, but it appears London rivals Chelsea have not reached a deal with the Yorkshire club.
MORE : Manchester United to sanction three player departures once club hire Tyrell Malacia from Feyenoord
MORE : Frenkie de Jong tells Manchester United he wants to stay at Barcelona
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