Ajeti has been widely expected to move on from Parkhead this summer and is firmly out of favour at the Scottish club.
A number of sides have looked at signing the former West Ham United man and he has agreed to go to Turkey.
Ajeti will join Gaziantep and, according to Turkish outlet Sports Digitale, will be in Turkey on Friday.
He is due to arrive in Turkey at 19:30 on Friday and will then put the finishing touches to his move to Gaziantep.
Ajeti is to join a club sitting bottom of the Turkish Super Lig after three games.
They have scored just twice and will be looking for Ajeti to give them more of a threat going forward.
Gaziantep are next in action on Saturday when they play host to Istanbul giants Galatasaray in the Super Lig.
