Semenyo's contract carries a £65m buyout clause, which can be triggered in the first two weeks of January.
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The Telegraph says United believe Manchester City are their closest rivals to land the Ghana international.
United manager Ruben Amorim met with Semenyo in London at the end of last season, but the board refused to meet the Cherries' £80m asking price.
City are also interested, though doubts over the future of manager Pep Guardiola may influence Semenyo's thinking.
Meanwhile, Amorim views Semenyo as an attacking wing-back in the mould of how Amad Diallo has thrived on the left-side of United's setup.