The Hurriyet daily said Podolski had agreed a three-year deal with the J1 side worth 15 million euros ($15.8 million) and Istanbul side Galatasaray would receive a transfer fee of 2.7 million euros ($2.85 million).
The move would take effect from June 1, meaning Podolski could play for Galatasaray until the end of the season, it said. The announcement and signing is expected in the next days.
The paper said that the player is enthusiastic about the move to Japan and Galatasaray were keen to accommodate him but insisted he stayed until the end of the season.
Vissel Kobe initially offered a transfer fee of 2.0 million euros but Galatasaray informed the Japanese they had paid his former club Arsenal 2.5 million for him.
“We are very happy with him. It’s not possible for us to sell him for a lower price,” Galatasaray told Vissel Kobe before the final transfer fee of 2.7 million euros was agreed, it said.
The reported move by Podolski comes with Galatasaray in a period of upheaval, with the club sacking manager Jan Olde Riekerink after one year in the job and set to appoint former Croatian international Igor Tudor as his replacement.
The German, who retired from international duty last year, joined Galatasaray from Arsenal in 2015 after a loan spell at Inter Milan.
Podolski’s tough approach to the game and clear interest in Turkish culture has made him popular with Galatasaray fans.
He has also endured injury problems and the club has been through a fallow period, finishing only sixth in the Super Lig last year and lagging in third this season.