McNair was one of Sunderland's brightest lights in a dismal campaign that saw them crash into the third tier of English football for only the second time in their history and looks set to depart the club this summer.
Brighton are thought to have two bids for the Northern Ireland international turned down already this summer, with the latter of those thought to be in the region of £3.5m.
However, Boro look to have blown those bids out of the water as Tony Pulis looks to revamp his Riverside engine room this summer.
Reports claim that the 'Teessiders' have offered a significant portion of the deal up front as they look for a way of tempting the cash-strapped 'Black Cats' into doing business.
McNair made 18 appearances for Sunderland last season, with his return of five goals not enough to save the club from relegation to Legaue One.