The England left-back, who returned from the World Cup with the rest of the squad on Wednesday, is still to undergo a medical but no problems are anticiapted - he will then go on holiday.
Shaw is one of United’s main summer transfer targets – they initially bid around £27m for the 18-year-old at the end of last season.
Veteran Patrice Evra signed a new one-year contract last month but United are keen to secure a long-term replacement for the Frenchman – especially after Alex Buttner agreed to join Dinamo Moscow earlier this week.
Shaw, who will become the world’s most expensive teenager if the deal goes ahead, has enjoyed a rapid rise after making his Southampton debut as a 16-year-old in January 2012.
He has made 60 appearances Premier League for Saints and made his England debut in a friendly against Denmark in March.
Roy Hodgson then selected Shaw ahead of Ashley Cole for England’s World Cup squad, although he only made one appearance, in their final game against Costa Rica.