Kahveci, 25, is a Turkey international and a star player in the Basaksehir side that has just won the Süper Lig.
The cultured midfielder would arrive as a replacement for Ever Banega, who is leaving for Saudi Arabia.
Banega’s exit frees up an international spot in Julen Lopetegui’s squad, with Kahveci in line to fill it.
Sevilla have already agreed the basis of a four-year contract with Kahveci’s agent.
Basaksehir president Goksel Gumusdag also confirmed to NTV Spor that Sevilla are ‘very interested’ in the player.
Kahveci made 38 appearances for the Istanbul club this season, scoring six and assisting five goals.