The England international has enjoyed a superb season in Germany, scoring eight goals and contributing 16 assists in 27 Bundesliga appearances.
Dortmund paid just £8 million to bring the 19-year-old over from Manchester City in 2017, but a breakthrough campaign this season means United will have to offer £100 million before add-ons to secure a deal, according to The Independent.
The report adds that Paris Saint-Germain and Real Madrid have also shown interest in the forward, but Old Trafford remains his most likely destination.
United last week confirmed Ole Gunnar Solskjaer as their permanent manager for next season and the Norwegian is believed to want to build a pacey, exuberant front line around Marcus Rashford, Anthony Martial and Sancho.
Alexis Sanchez, who joined the club from Arsenal in January 2018 but has struggled to settle, could be sold in the summer window.
The Bundesliga leaders currently have Sancho tied down for another three years on his contract, which is why a nine-figure sum will be required.
Manchester City would need to be informed of any agreement due to the ‘matching clause’ they agreed when he was sold, but this is said to be unlikely to impact United's prospects of signing the winger.
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