The 20-year-old Kagawa, whose contract with Cerezo Osaka concludes this season, is also being asked to remain with the J-League second- division club.
Cerezo could win promotion to the top flight and talks with Kagawa are likely to begin if they are promoted.
This season, Kagawa has netted 27 goals in 42 matches to lead the second division in scoring. He left the national team to return to Cerezo on June 8.