The club's owner Hans-Joachim Watzke revealed the German champions want a 'princely sum' despite the 23-year-old having just a year remaining on his contract.
Kagawa, who netted 17 goals for Dortmund this season to help them secure a second consecutive title, has already met for talks with Manchester United boss Alex Ferguson, while he has also been linked to other high-profile clubs including Arsenal, Chelsea and AC Milan.
But Watzke told Sport Bild that the club would not be fooled into selling him for a derisory price.
"We all know the abilities of Shinji, his youth and the value he adds to this side," said Watzke.
"I also note what Bayern Munich paid for goalkeeper Manuel Neuer (22 million euros from Schalke last year).
"He also only had a year left on his contract. Given those facts we can certainly expect a princely sum for him."