The Daily Express says United have lodged a bid of £27million for the England star, who was also wanted by Chelsea.
But according to reports, Blues boss Jose Mourinho pulled out of the running after refusing to match United's contract offer.
It is claimed Shaw has agreed in principle a £160,000-a-week deal at Old Trafford - which would make the 18-year-old the highest-paid defender in world football.
The entire package could cost United an astonishing £40m, with Shaw expected to sign a five-year deal.
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