The Georgia winger is a target for Newcastle and Manchester City.
But Ananidz told Sport-Express: "I am aware of the options he has. Yes, these are top English clubs. But I think he needs to stay at Napoli for at least another year: win the Italian championship, prove it to everyone next season, and then go to one of the strongest clubs in the world."
According to Ananidze, Kvaratskhelia is capable of becoming the best player in the history of Georgian football.
"He has everything for this," he added. "So far, this unspoken title is held by Kakha Kaladze. But Khvicha is quite capable of overtaking him."