Toure was a £16 million arrival from Arsenal three years ago and remains one of the most popular players at City.
However, he has never recovered his first-team spot after being forced to spend six months on the sidelines following a failed drugs test, which was blamed on the 31-year-old taking one of his wife's slimming products.
"We have been in talks with Kolo Toure over the past two days - it would be one of the most important transfers to be completed by an Anatolian side," Bursaspor manager Ertugrul Saglam told Dogan News Agency.
"Obviously nothing is certain until he signs pen to paper, however we are working hard to try finalise a deal."
Toure did made enough appearances last season to qualify for a Premier League title winner's medal last term as City ended their 44-year wait for the championship.