The former Manchester United manager, who is believed to have several job offers on the continent, is set to hold a meeting with the board at the Turkish club on Friday
Moyes was pictured at Istanbul Ataturk Airport on Friday and is understood to be in Turkey to hold a meeting with the club’s board and visit their training facilities.
The former Manchester United boss, who has been without a club following his sacking at Old Trafford on April 22, is believed to have a number of job offers from clubs in Europe.
The Scot could now replace former Manchester City boss Roberto Mancini in Istanbul after the Italian was sacked by the Turkish club earlier in June.
Moyes has made no secret of his desire to return to football management at the very top level.
"I got my first real taste of [the] Champions League this year and I thought I showed I can manage at that level," he told The Guardian earlier this month.
"I would like somewhere that would give me a chance of being in the Champions League or a club that has ambitions of being in the Champions League.
"It could be at home or abroad. I have always liked the thought of broadening my horizons and learning a bit more and at the moment I am quite comfortable watching and learning.
"I will go away and learn and look again to see what is happening."
Galatasaray will enter the Champions League at the group stage this season despite finishing second in the Turkish Super Lig due to Fenerbahce's ban from European football.