The Red Devils have been heavily linked with the 21-year-old for a second summer transfer window in a row, with the latest reports claiming that they have returned with an improved offer after seeing their initial approach turned down.
Man United officials are understood to be confident of bringing Sancho to the club this summer, despite Dortmund so far refusing to budge from their asking price of more than £80m.
Personal terms and a five-year contract are thought to have already been agreed, but German outlet Ruhr Nachrichten reports that Dortmund have now set Man United a deadline of the middle of July to come to terms over a transfer fee.
BVB employed similar tactics in last year's negotiations, with Man United ultimately missing the deadline they set out on that occasion and subsequently missing out on the player.
Sancho is thought to be keen on a return to England, having come through the ranks at Manchester City before joining Dortmund in 2017.