The Turkish club confirmed they have spoken to Moyes but insist the Scot is not top of their managerial shortlist.
Moyes has been in Instanbul to meet club president Unal Aysal, but Galatasaray insists Joachim Low and Jurgen Klinsmann - national coaches of Germany and the USA respectively - are their top targets.
A Galatasary spokesman said: "The president has given a speech about it.
"He said that we just spoke with David Moyes and he said that he can come to Galatasaray but he is not one of the main targets for us.
"They just spoke to each other, exchanging ideas - that it is it. It is true the president talked with David Moyes.
"We are not in a rush, we are waiting for the end of the World Cup. The first two targets are Low or Klinsmann."
Galatasaray parted company with Italian coach Roberto Mancini a fortnight ago after they finished second in the Turkish Super Lig.
Former Manchester City boss Mancini had been in charge since October 2013 and won the Turkish Cup last season.
Moyes has been out of work since being sacked by United in April after just 10 months in charge at Old Trafford.
The former Everton boss, who has been linked with a number of vacancies since his departure, has refused to rule out taking a top appointment abroad.
@bludsucker That's just you falling for his propaganda, Moyes is a mid table manager, no more, no less, he could have 10 Messis and never win anything, he just isn't good enough. Woodward wasn't the most on the ball last summer, but I don't accept that all the teams above United had better squads than United's, the skill in being a manager is getting players to play to their potential, United's entire squad went backwards under him, he can't blame Fergie, Woodward or anyone else for his own failings. If a Liverpool fan had taken over from Fergie things wouldn't have gone any worse.- solskjaer99