The Turkish club said in a statement on Tuesday that the 28-year-old would cost a transfer fee of 3.5 million euros.
Eboue must wait for his debut after the start of the Turkish season was put back to September 9 due to an ongoing investigation into league match-fixing allegations.
Galatasaray are not in Europe this season after they finished eighth in the top-flight last term, 36 points behind champions Fenerbahce.
Eboue, who joined Arsenal from Belgian club Beveren in January 2005, can play in defence or midfield.
He made 214 appearances for the London team and was an integral part of the defence which kept a record 10 Champions League clean sheets en route to the 2006 final in Paris where they lost 2-1 to Barcelona.
Eboue featured in the 2010 World Cup in South Africa.
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