The 23-year-old has been in stellar form this season and his 12 goals in 25 league games have helped put defending champions Dortmund on top of the German league with six games remaining.
But according to Bild, Kagawa is considering rejecting Dortmund's offer to double his wages from 1.5 million to three million euros per season and extend his contract, which is due to expire in June 2013, until 2016.
Both Manchester United and Chelsea are reported to be interested in signing the Japan star and reports in the German media claim Kagawa is worth around 15 million euros.
"There will be a decision made soon," said Dortmund manager Michael Zorc.
"We would love to extend to his contract, but the ball is in his court.
"We know that he fancies a move to England or Spain.
"The question is at what point in time."
Kagawa has been one of the German league's star finds, having signed from Cerezo Osaka in 2010 for 350,000 euros and scored two goals leading Dortmund's attack in their 6-1 thrashing of Cologne 10 days ago.