The Bundesliga champions are determined to hold on to the Japan international, and have reportedly offered the attacking midfielder a new deal worth €3m per year
The 23-year-old currently earns €1.5 million per season, but BVB have offered the Japan international a new deal worth €3m per year.
Kagawa's current contract runs until the summer of 2013, but Dortmund are keen to keep him at the club until June 2016.
"Shinji is one of our key players and we are very keen to extend with him. The ball is in his court though," sporting director Michael Zorc commented to Bild.
Kagawa joined Borussia Dortmund from Cerezo Osaka in the summer of 2010 and has since developed into a key player at BVB, who earlier this week tied down Mario Gotze until 2016.
The Japan international has netted 11 times in 24 Bundesliga appearances this term, as well as setting up seven more.