The Red Devils are putting the Everton striker through the tests during his holiday in Los Angeles.
Starsport understand that United have agreed a £75m fee with Everton for Lukaku.
The Belgium international has given the green light to a long-term contract at Old Trafford worth £250,000-a-week.
Lukaku has been holidaying in the United States of America with United midfielder Paul Pogba.
Pogba has persuaded Lukaku to snub a return to Chelsea and join him in Manchester, according to the Mirror.
United hope they can complete the transfer before United begin their pre-season tour of LA on Sunday.
The Mirror report that Jose Mourinho hopes Lukaku will be able to train with his new team-mates on Monday.
Chelsea believed they were set to sign Lukaku, who was flogged to Everton in 2014 for £28m.
But Starsport exclusively revealed in May that United manager Jose Mourinho wanted to land Lukaku.
And United have been working to beat their bitter rivals to the transfer for a number of months.
As a result of agreeing a fee with Everton, the Red Devils have now dropped their interest in Real Madrid striker Alvaro Morata.
Morata was heavily linked with a move to Old Trafford but could now emerge as a target for Chelsea.