Dortmund are of the belief that the deal to take Sancho to United will be finalised by the end of the week, says The Times.
It comes as the Red Devils looks to ramp up their efforts to sign the England international.
Ole Gunnar Solskjaer has made the forward his top target after failing to agree a transfer last summer following a breakdown in talks.
Dortmund priced Sancho at £109m, a target that was too high for the Red Devils, but that valuation has dropped dramatically this summer.
They now want a fee of around £85m, though United are still playing hard-ball and have offered around £10m less than that.
Nevertheless, there remains a cautious optimism that a deal will go ahead.