Smith is currently on holiday in the States but Southgate is determined to sell Middlesbrough to the 25-year-old England international.
Southgate's Atlantic crossing puts Boro a step ahead of north east rivals Newcastle and Sunderland, both of whom have been tracking the former Leeds star.
Neither Sam Allardyce nor Roy Keane have lodged bids with Old Trafford, leaving Southgate in the box seat.