talkSPORT reported on Wednesday that the two clubs were in talks to agree a fee for the 24-year-old, which they are now understood to have done.
Man United have already had a £42million bid for the Ajax man rejected, but are now said to have met the Dutch club's £46m asking price.
Journalist Fabrizio Romano said the final step is deciding on the payment terms, adding the deal is 'imminent.'
The Premier League side's manager, Ten Hag knows Martinez well following his time as Ajax boss before he left for Manchester this summer.
And the Dutchman was clearly keen to reunite with the versatile defender, who made 36 starts in all competitions under him last season.
The Argentine played a key role in the club's back-to-back Eredivisie title wins, where he played in a number of positions, including centre-back, left-back, and centre-midfield.
The completion of the deal will cap a good week for the Red Devils after their 4-0 pre-season win over Liverpool in Bangkok.
Jadon Sancho, Fred, Anthony Martial and Facundo Pellistri scored in Ten Hag's first game in charge against their rivals, who made a number of changes to their side.
Despite the comfortable win, Man United boss insists that they still need to strengthen both their midfield and attack.
"We want to have the right players," Ten Hag said this week.
"We have a really good squad, we are working on developing our way of playing and if the opportunity is there we will strike [to sign new players].
"I said already definitely we are searching for players in midfield and I think that we have a need in offence. We are still working for players."
That could include Martinez's Ajax teammate Anthony and Barcelona midfielder Frenkie de Jong.