The Argentinian has been the centre of transfer speculation throughout the transfer window, with Arsenal and United battling it out to sign the centre-back.
However, a recent report suggested that Arsenal had cooled their interest in Martinez, while the Red Devils held a meeting with Ajax representatives on Wednesday to discuss a potential transfer.
According to transfer expert Fabrizio Romano, talks in Amsterdam were "positive" and "successful", with the transfer believed to be imminent.
The two clubs are close to agreeing on a fee in excess of €50m (£42.3m) ahead of the final round of talks, with United hoping to confirm the deal in the next two days.
Martinez is set to depart Ajax after making 118 appearances in a three-year spell with the club.