The newspaper states that Old Trafford is the ‘most likely destination’ for the 24-year-old.
The report goes onto state that Dortmund want a fee in the region of €25 million for their in-demand midfielder.
Bayern Munich and Barcelona have both been closely linked with Gündoğan in recent weeks.
Dortmund CEO Hans Joachim Watzke has previously confirmed that Gündoğan will leave the club at the end of the season.
“Gündoğan will be leaving Borussia Dortmund this summer, that is a fact. I am very confident we will see various clubs coming in for him,” Watzke told SportBild.
Gündoğan has five assists and three goals in 32 appearances for Dortmund despite spending much of the season out injured and is almost certain to leave the Bundesliga giants over the summer.