"I've healed to a point where I'm running again," Kagawa, who underwent surgery two months ago, told Kyodo news agency before leaving from Osaka's Kansai international airport.
"Ideally, I want to come back in mid to late-April. I want to head back and rejoin the team as soon as I can. I want to help out the team in any way I can so we can win the championship," the 22-year-old said.
Table-topping Dortmund lead Bayer Leverkusen by seven points in the Bundesliga race, chasing their first title since 2002.
Kagawa broke a metatarsal in his right foot during Japan's semi-final victory over South Korea at the Asian Cup in January.
He underwent surgery in Tokyo on 30 January and has been recovering in Japan.
He moved to Dortmund from J-League side Cerezo Osaka in July and scored eight goals in 17 Bundesliga games to help Dortmund's charge to the top of the table before the winter break.
With four Europa League goals to his credit as well, Kagawa had reportedly come onto the radar of Spanish clubs Villarreal and Atletico Madrid, as well as Juventus and Tottenham, before his injury.
He missed last year's World Cup as then Japan coach Takeshi Okada made him only a non-playing support member of his squad in South Africa.