The Sun says Forest see Sancho as a replacement for Anthony Elanga, a former United player, who has just been sold to Newcastle United for £55m.
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Sancho, despite his reputation, is available from United for around half Elanga's transfer fee.
Forest owner Evangelos Marinakis is prepared to fund a move for Sancho, who spent the second-half of last season on-loan with Chelsea, though didn't do enough to convince the Blues to sign him permanently.
Sancho has been the subject of interest from across Europe, though for now no serious offer has been lodged with United.