The German club have told United they want in excess of £100m for the 20-year-old England winger.
There is no agreement yet on a fee, which would be paid in instalments.
Dortmund have set the deadline because they want undisturbed preparation for the 2020-21 Bundesliga season, reports BBC Sport.
The new campaign starts on 18 September, with the transfer deadline for international deals on 5 October.
Sancho has been at Dortmund for three years, having signed from Manchester City in August 2017 for a fee of about £10m.