If the deal goes through the 18-year-old left back will become the most expensive teenager ever, beating the £25m fee United paid for Wayne Rooney in 2004.
United had a £27m bid rejected by Southampton last month and it is believed new manager Ronald Koeman is keen to keep the talented youngster at St Mary’s.
Shaw is set to make his competitive debut for England tonight, starting in place of the injured Leighton Baines against Costa Rica.
Southampton are not afraid to demand big fees for the talented youngsters that come through their ranks.
Alex Oxlade-Chamberlain cost Arsenal £15m three years ago and Liverpool are currently in two minds after being told by the club that Adam Lallana would cost £30m.
New Manchester United boss Louis van Gaal is determined to land the youngster and has told United chief Ed Woodward to offer the teenager £100,000-per-week.
United vice-captain Patrice Evra, 33, will remain at the club for one more year but understudy Alex Buttner has already left for Dinamo Moscow.
The Red Devils are also close to completing a £29m deal for Ander Herrera, while former defensive stalwart Rio Ferdinand is set to move to QPR.
England manager Roy Hodgson said tonight’s game against Costa Rica will be a good chance for Shaw and the other youngsters to impress.
“I’m looking for them to show their talent,” said Hodgson. “Both of them are very good passers of the ball, very good at running with the ball.
“Both, for young players, have good vision for the game. That will improve the more games they play, but they’re well advanced for their age. This is a chance for them to show they are good players.
“They have an opportunity to play from the start, and together in a game with other young players around them… they can show the faith and hope we have that those boys are ready to make their mark.
“This could be the first moment they give us an inkling that we are on the right track.”