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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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.