The Japan international joined Gala in January on-loan to June.
“I want to play and remain at Galatasaray," the 31-year-old Japan international told Milliyet.
“I have settled in well and I have been reborn here. I like to play football and I knew what the club was like. I have immediately settled in to Istanbul."
It's reported Galatasaray are prepared to pay Inter €2.5m to buy Nagatomo outright and raise his current salary of €850,000.