The 34-year-old, who broke the British transfer record when joining City from Real Madrid for £32.5m in 2008, arrives from fellow Turkish club Sivasspor.
He will join former City team-mate Emmanuel Adebayor at Basaksehir, plus Gael Clichy, who played at the Etihad Stadium after Robinho left in 2010.
Basaksehir were knocked out of the Europa League by Burnley in August.
Robinho, who began his career in his native Brazil with Santos, spent five years with AC Milan after leaving City, before playing in China for Guangzhou Evergrande and returning to his home country to join Atletico Mineiro.
In November 2017, he was sentenced in his absence by an Italian court to nine years in prison after being convicted, along with five other Brazilians, of the 2013 gang rape of an Albanian woman in a Milan nightclub.
Robinho, who pleaded not guilty via his lawyer during the case and subsequently published an Instagram post denying any wrongdoing, said he would contest the sentence.