The 30-year-old is out of contract at the end of the season and Liverpool have not offered a new deal.
Johnson started his professional career at West Ham before moving to Chelsea in July 2013.
According to Turkish newspaper Fanatik, Johnson is open to a move abroad and Galatasaray are keen on securing a deal.
“I expect to leave at the end of the season, I have no contract,” said Johnson.
“I don’t think that will change but we have to wait and see. I have always been professional, always wanted to play and that’s still what I’m doing.
“I’ve got four more games and then let’s see what we do in the summer.”
Galatasaray are planning to strengthen their right-back position over the summer.
The Istanbul giants are expected to part ways with right-back Emmanuel Eboue at the end of the season. The experienced right-back has been left out of the team all season.
The Lions have been fielding former reserve team player Sabri Sarıoğlu as right-back and £4 million summer signing Tarık Çamdal has failed to impress.