The Daily Star reports that United boss David Moyes is keen to sign the 18-year-old defender in January.
Southampton are keen to keep the England Under-21 defender at the club and have put a huge price-tag on the youngster.
United face competition from Chelsea and Tottenham in the race to sign Shaw.
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Liverpool have unveiled their new home kit for next season, sticking with the hugely popular retro styling of the current shirt.
The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.