The Bundesliga side are thought to have demanded €120m up front but are now reportedly willing to accept the lower amount, as long as add-ons can take the fee up to the desired amount, which would make Sancho the most expensive Englishman in history.
His performances over the course have piqued the interested of United, who want to bring the player back to Manchester, after he left rivals City for Germany at the age of 17.
Barcelona and Real Madrid have been mentioned as potential suitors in the past, but Man United have always been his keenest admirers, with Ole Gunnar Solskjaer looking to add more young, exciting English talent to his squad.
Even though the Old Trafford side have been interested in Sancho for some time, talks with Dortmund and the player have intensified over the last week after they secured Champions League football.
Should the transfer go through, it will be the second time in a year in which Man Utd have broken the transfer record for an English player, having signed Harry Maguire from Leicester for €94m last summer.