The England Under-21 pair are both on Teesside undergoing medicals and discussing personal terms.
Both players had been offered new deals to stay at Old Trafford by Sir Alex Ferguson, but they have decided that they must move to gain regular first-team football.
22-year-old Wilson has never actually started for United`s first team, while Greening, also 22, has made five starts for The Red Devils.
Boro boss Steve McClaren moved for the duo last week and has fended off interest from a number of clubs to sign both players, who will become Boro`s third and fourth captures of the summer after Gareth Southgate and Szilard Nemeth.
The pair are likely to be unveiled on Sunday when Boro entertain Spanish side Athletic Bilbao in Curtis Fleming`s testimonial.
