Moyes was sacked after less than a year in charge of Manchester United, and has been out of work since Easter.
The Turkish club said they had held an "exchange of ideas" with Moyes after he was spotted in the country.
Many experts suggested the Scot, who has also been in charge at Preston North End and Everton, would be better off making his next job overseas after his torrid time at Old Trafford.
Roberto Mancini, formerly of Manchester City, was last in charge of Turkish Super Lig side Galatasaray but parted company with the club earlier this month despite guiding them to the Champions League last-16.
INSIGHT FROM TURKEY
We asked Aslan Alper from our Turkish site to gauge the reaction to news of Moyes' possible appointment...
"Galatasaray fans are shocking now. They don't want Moyes, because Moyes has such a poor record at Manchester United. They are thinking it is the wrong choice after Fatih Terim and Roberto Mancini. I think Moyes wont see end out the season in Turkey - Turkish fans and football directors don't like long term plans."
- Sports & Recreation
- Manchester United
- Roberto Mancini
- Turkish Super Lig