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.