The 35-year-old striker spent almost three years in Turkey after joining Basaksehir in January 2017, playing for the Istanbul club for two-and-a-half seasons before signing for Kayserispor in August 2019.
He scored two league goals in eight league appearances this term including one in his final game - a 4-1 defeat at Besiktas on Monday.
After the game, he announced his departure, telling reporters: "These are tough decisions and [it] was tough for me. One of the most important decisions of my life.
"I tried to give my energy and will to this team since I arrived. I gave my best but sometimes you have some issues in your personal life.
"You think about your family and have to make decisions about your future. And that decision should be fast. I have decided not to continue my career here. Even though I am leaving physically, my heart will always beat for this team."
Adebayor has strong links with English football, having scored top-flight goals for Arsenal, Manchester City, Tottenham and Crystal Palace.
His most prolific season in England came in 2007-08 when he scored 24 league goals in 36 Premier League appearances for Arsenal, and he did not rule out a move away from Turkey as he seeks a new club.
"I cannot say that I'm leaving Turkey for good," said Adebayor. "Anything can happen in life.
"I have to rest and spend some time with my family. I had three great years here and I think that the Turkish league is among the best leagues in Europe."