Sancho, 20, stood out last season with 17 league goals and as many assists as Dortmund finished second to Bayern Munich in the Bundesliga.
He scored the first hat-trick of his career in a 6-1 drubbing of bottom side Paderborn at the end of May.
The fleet-footed forward, who joined from Manchester City’s academy in 2017, has repeatedly been linked to their rivals United and Dortmund want 120 million euros for him ($136 million).
A decision whether Sancho stays or goes is expected at the end of August.
So far, Dortmund have signed Belgium international Thomas Meunier at right-back on a free transfer from Paris Saint-Germain to replace Achraf Hakimi.
The German giants are also expected to sign English youngster Jude Bellingham, 17, from Birmingham City for around 25 million euros before next term.
