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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The reputation of the Premier League means it is the most popular and one of the most watched leagues in the world.
Five English clubs will compete in the last sixteen of the Champions League for the first time ever as Liverpool, Manchester United, Manchester City, Chelsea and Tottenham face some difficult away matches.