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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Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.
Newcastle United are staring down the barrel of relegation to the Championship for the second time in the past eight years.