The club views Sancho as a suitable replacement for Anthony Elanga, who has recently been sold to Newcastle United for £55 million.
Sancho, despite his high profile, can be acquired by United for a transfer fee roughly half of Elanga's price tag.
Forest's owner, Evangelos Marinakis, is willing to fund the move for Sancho, who spent the second half of last season on loan with Chelsea but failed to impress the team enough to secure a permanent deal.
Sancho has attracted interest from various European clubs, but at present, no serious offer has been made to Manchester United.
