The Ghana international is being linked with England's biggest clubs, including Arsenal and Liverpool, this season.
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The Daily Mail says United are also interested in the free-scoring attacker, though Bournemouth's valuation could turn them off.
Given the frenzied interest, Bournemouth are demanding a starting price of £70m for a summer auction.
Semenyo has scored 10 goals and provided six assists in all competitions. Any move from United will hinge on major sales, including Aston Villa loanee Marcus Rashford.