The 29-year-old, who joined Arsenal from Fenerbahce in 2011, has made just two Premier League appearances for the Gunners this season.
England international Kieran Gibbs has established himself as Arsenal's first-choice left-back this season, leading to Santos questioning his position at the club.
"I would like to state clearly that I have a lot of respect for Fenerbache, and I love Istanbul... but for now I am a Gunner," Santos tweeted.
"For now, I have not received an offer from a Turkish team. However, if I receive an offer one day, thinking what is best for my family, I will make a decision."
Santos has 22 Brazilian caps to his name.