Shaw has been widely mooted as a target for United this summer and it appears they are keen to get their business done before the World Cup and before they have appointed a permanent manager.
They ended their season, with Ryan Giggs in interim charge, against the Saints on Sunday at St Mary's but there is no official confirmation from either side that any talks about Shaw have taken place.
Shaw has made a massive impact since his Southampton debut in November 2012, quickly nailing down a regular role and doing enough to earn his senior England debut against Denmark in February.
He is nervously waiting to find out whether he will be selected by Roy Hodgson on Monday for his World Cup squad and might soon have a decision to make about his club future.
United are keen to sign a young left-back but will have to find a huge transfer fee to tempt Southampton, with Shaw having signed a new five-year contract on his 18th birthday last July.
Their veteran left-back Patrice Evra is out of contract this summer and could have played his last game for the club on Sunday.
United could also appoint their next permanent manager in the next week, with Netherlands national coach Louis van Gaal still favourite to take over from David Moyes.