Arsenal have been chasing the Argentina defender all summer but have failed with two bids for the Ajax star.
But Manchester United are now pushing to hijack their pursuit of the centre-back and have gone in with a bid, believed to be in the region of €43m to €45m.
Ajax are yet to respond to the offer but negotiations continue between the two clubs and more talks are expected to take place over the weekend.
Manchester United are hoping to establish the final price for Martinez in the next round of talks with Ajax.
Arsenal are still in the race and are not expecting problems over agreeing on personal terms with the defender.
However, Manchester United feel that they have the edge in the race over Arsenal due to Erik ten Hag.
Martinez enjoyed playing under the Dutchman in the last three years and is believed to be keen on working with him again.
Manchester United are aiming to finalise the arrivals of Tyrell Malacia and Frenkie de Jong before they fly out for their pre-season tour on Friday.