Traore joined Villa under then boss Dean Smith two years ago but struggled to nail down a regular first team place due to periods of injury and loss of form.
The 26-year-old has scored seven times in 45 appearances for Villa since joining in a £17million switch from French giants Lyon.
However, he has failed to impress new boss Steven Gerrard and appears surplus to requirements at Villa Park.
Basaksehir finished fourth in the Turkish Super Lig last season.
Reports suggest a switch could be on the basis of an initial loan with an option to buy.
The former Chelsea attacker had also been linked with a possible move back to Lyon during the current transfer window.