The Daily Star reports that United boss David Moyes is considering making a January bid for the Southampton defender.
But Shaw could decide to turn down the chance to join United as he dreams of joining Chelsea.
A Saints insider told the Daily Star: “Luke’s a huge Chelsea fan and he’d be delighted to play for them.
“United’s interest is flattering but he knows he doesn’t have to rush anywhere just yet.”
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Summer 2016 is set to be a very exciting time for football fans, as this is when the long awaited Euro 2016 tournament kicks off.
Since Guus Hiddink arrived at the helm, Chelsea have improved considerably after a tricky start to the campaign.