The report claims that Blues boss Jose Mourinho could be tempted to let Costa leave on a temporary move following a public spat with the striker.
Costa is of course no stranger to transfer talks with Turkish clubs as he almost joined Beşiktaş in 2011 before being sent out on loan to Rayo Vallecano.
The Spanish international told Turkish-football.com: “I had agreed terms with Beşiktaş and was going to join however, I suffered an injury and the club pulled out of the move.”
Costa has four goals and three assists in 18 appearances for the Blues this season but has another four years left on his deal.
Galatasaray are looking to boost their forward line over the winter transfer window and have been keeping close tabs on a number of forwards.
The Istanbul giants are the reigning champions but are currently five points behind league leaders Beşiktaş and recently sacked double winning manager Hamza Hamzaoğlu.
The former Turkish national team assistant coach was replaced by Galatasaray legend Mustafa Denizli a fortnight ago.