Ghana international is being tracked by the Reds, though management had planned not to move for him until the summer.
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However, a major injury blow to Alexander Isak has directors rethinking that strategy.
Isak suffered a "lower leg injury" in victory at Tottenham on Saturday, leaving management "very concerned".
As such, the Telegraph says a move for Semenyo is being discussed ahead of the January market opening inside a fortnight.
Semenyo's deal carries a £65m buyout clause which can be triggered in the first two weeks of the winter transfer window.