The Red Devils have lodged two offers for Sancho, the latest of which was worth around £75m.
That amount is close to Dortmund's valuation, and Sky Sports says both clubs expect to complete the deal.
Another stickler has been the the structure of payments as Dortmund prefer the majority of the fee paid up front with the rest in achievable add-ons.
But United have avoided another hurdle as they agreed personal terms with Sancho last summer.
Dortmund want a resolution before they return for preseason training camp on July 23.
Sancho is under contract with Dortmund until the summer of 2023.