Moyes is targeting both Saints stars, who impressed in England's friendly win over Denmark.
The Daily Express says Moyes is reluctant to offer Southampton more than £30million for the duo, but the south-coast club will not budge for anything less than £40million.
Shaw is set to be involved in an all-out bidding war this summer as Chelsea and Manchester City prepare offers for England's latest star, but Moyes feels the offer of first-team football at Old Trafford could lure the youngster to Carrington.
On Lallana, Moyes believes his versatility and creativity can rejuvenate United's stale midfield.
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