The 19-year old centre-half has penned a five-year deal with the English champions with the transfer fee believed to be £16.5million.
Jones had agreed personal terms with United last week but the deal appeared to have hit a snag with Blackburn holding out for more money.
But it is now close to being complete after the Red Devils released a statement this afternoon.
"The player passed a medical in Manchester last week and has agreed a five-year contract.
"The deal will be completed once the player returns from international duty (with the England U21s)."