Bayern are preparing a £9 million bid for the Turkey international, who is being seen as a potential replacement for Franck Ribery.
Juve are also keen, but will need to convince Turan, who said: "I was born in Istanbul and I have always been a fan of Galatasaray.
"I grew up in the youth team and, apart from a loan to Vestel Manisapor, I have always worn these colors. My dream is to end my career here.
"I would only leave if the club tells me that my departure is necessary to balance the company accounts."