The 26-year-old centre-forward joined Celtic from West Ham United in the summer of 2020 on a big-money move.
However, Ajeti has failed to justify his price tag and he spent last season away from Celtic Park on loan at Sturm Graz.
Celtic are keen to offload him this summer and the forward has received interest from Turkish Super Lig clubs.
Gaziantep are particularly keen on landing Ajeti this summer and Trabzonspor also joined the race for the Celtic star.
However, it has been claimed that Gaziantep have now convinced the 26-year-old to join them.
And the Turkish outfit are set to pay £500,000 to Celtic to acquire the services of Ajeti, who is in the final year of his contract at Parkhead.
Ajeti has featured a total of 48 times for Celtic in his career, scoring nine goals while laying on five assists.