Zaha, 30, came up through Palace's academy and has spent the majority of his career at the club, apart from a two-year spell with Manchester United.
He has played over 450 games for Palace, scoring 90 goals.
He has also been capped 31 times by the Ivory Coast, having made his debut in 2017. He played for England at youth level and made two appearances for their senior side before switching his international allegiance.
"Official negotiations have started with professional footballer Dazet Wilfried Armel Zaha regarding the transfer of the player to our club," Galatasaray said in a statement on social media.
Galatasaray claimed a record-extending 23rd league title in May, their first since the 2018-19 season.