The Bundesliga club’s sporting director Michael Zorc has issued the stark warning to the Red Devils.
And Ole Gunnar Solskjaer and co. will have to act quickly if they are to tie up a deal for England international Sancho.
It was reported when the transfer window opened on Monday 27 July that Dortmund had rebuffed an £89m from United.
They are yet to come back with an improved offer - leading Zorc to lay out a plan for when he wants to wrap up his dealings in the transfer market, as well as a non-negotiable asking price.
He initially labelled the gossip as "purely a media topic".
But now Zorc has given a specific date of August 10 that he wants the transfer business regarding Sancho to be done by.
“There is no room for negotiation, 120 million Euros is the only acceptable fee. Dortmund start their training camp in Bad Ragaz, Switzerland, on Aug 10, there will be no sale after that date.”
United are set to splash the cash this summer after chief executive Ed Woodward insisted the club were in a healthy position financially, despite the impacts of the ongoing pandemic.
"The ongoing disruption from the pandemic – and the economic pressures it has created – has shown why it is so important for football clubs to focus on their financial stability,” he said.
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"We are fortunate to have a strong and resilient commercial business which has supported long-term investment in our team.
“That will continue, albeit with extra caution in the coming months as we navigate our way through this uncertain period."