The 21-year-old broke a bone in his foot in Japan's semi-final win over South Korea on Tuesday, which ended in a 3-0 penalty shoot-out victory to the Japanese following a 2-2 draw after extra-time.
Kagawa broke down in training on Wednesday and a scan revealed the extent of the injury.
Bundesliga leaders Dortmund confirmed he will miss Saturday's final against Australia in Qatar and the injury threatens to end his season prematurely.
"The news is bad," Dortmund's press officer Hans-Joachim Watzke told SID, an AFP subsidiary.
"He was a prominent figure for us in the first half of the season.
"The season (in Germany) is three-and-a-half months away and a broken foot needs time to heal.
"It will be close (whether he plays again this season)."
The attacking midfielder has become a star in the German league since arriving from Cerezo Osaka in June 2010.
He scored eight goals in 17 Bundesliga matches for Dortmund to make him the club's current joint top-scorer.