According to Sky Sport, the Red Devils proposed a €100m deal – €80m plus €20m in bonuses.
The offer was quickly rejected by Dortmund, who have stuck to their €120m valuation for the England forward.
England international Sancho, 20, has been and remains United’s number one target this summer.
Six weeks ago, Dortmund chief Michael Zorc said there was no chance of Sancho leaving this summer.
Several BVB players have since given interviews stating that Sancho will stay for another year.
But United are still pushing for a deal before the end of the transfer window and could return with a higher bid.
Sky Sport claims that everything besides a fee (wages, commissions, etc.) have been agreed.
But unless United make Dortmund an offer they can’t refuse before the deadline on October 5, Sancho will remain in Germany for the season.