The Premier League champions have edged ahead of Chelsea in the battle to capture the Southampton left-back, who is expected to leave St Mary's this summer
The Red Devils have edged ahead of Chelsea in the battle to capture one of the highest-rated youngsters in English football.
United are prepared to spend upwards of €30m to capture Shaw - even if Patrice Evra accepts the club's offer to extend his contract beyond this season.
As reported by Goal, United opened preliminary talks over a move for Shaw in January and have stepped up their interest in recent months through regular dialogue with Stellar Group, the agency who represent the newly-capped England international.
United are set to press ahead with a bid for Shaw even though Louis van Gaal will shortly be confirmed as the new manager in succession to David Moyes, who had identified the left-back as a key target and had done a lot of the groundwork for a deal.
Chelsea have also registered their interest in the 18-year-old, a boyhood Blues fan whose parents live in the south-west London suburb of Raynes Park, but boss Jose Mourinho has indicated he is not prepared to make Shaw a regular starter next season.
Liverpool, Manchester City and Tottenham are also monitoring the youngster's situation but they are not currently as advanced in the race for Shaw's signature as United and Chelsea.
There is growing confidence at United that they can capture the teenager, who is expected to be named in England's World Cup squad next Monday. United would like a deal in place before the World Cup, as they are aware that his value could rise if Leighton Baines' expected deputy shines in Brazil.
Old Trafford officials want to keep Evra for at least two more seasons and allow him to mentor Shaw while he gradually establishes himself in the season. The Frenchman, who is being coveted by Monaco and Inter Milan, has said he will make a decision on his future after this weekend's Premier League finale.
Shaw has had an outstanding 2013-14 campaign for Southampton and was recently named in the PFA Premier League team of the season.