The 30-year-old Ivory Coast international is a free agent after his Palace contract expired in June.
Palace reportedly offered Zaha £200,000 a week to stay at the club where he spent most of his career.
Turkish champions Galatasaray, who will be in the Champions League this season, said “formal negotiations have been initiated” with Zaha.
Zaha later said in an Instagram post that he would not be returning to Palace.
“As I start a new chapter I want to speak to you Palace fans who mean so much to me,” he wrote.
“We have been through so much together, including 10 years in the Prem [Premier League]. There are loads of amazing memories that will stay in my head forever and also some really tough times along the way, but I’m so proud to have played my part in getting us to the Prem and helping to keep us there.
“I’ve been wearing the Palace shirt since I was eight years old, it’s literally been my second skin and I always gave everything when I wore it.
“You have all seen me angry and frustrated, but you’ve also seen my hunger and passion and the joy that I take from winning, and I hope that is the memory that lives on in the minds of Palace fans.”