Formula One raced at the Hermann Tilke-penned Istanbul Park - featuring the famous four-apex Turn 8 corner - between 2005 and 2011.
Bernie Ecclestone and race organisers could not agree a new race fee for 2012 and beyond.
But late last week, British journalist Bob Mackenzie, of the Daily Express, revealed that the F1 chief executive is "pushing for a return" of the Turkish event, "after ruling out a French race".
Greek website gocar.gr said the decision will be made in the coming week.
Turkey's change of fortune is due to a local businessman, Vural Ak, buying the circuit's management rights.
"I've known Bernie Ecclestone for many years," he said.
Ak, who is willing to pay $5m out of his own pocket, said Ecclestone is asking for a $26 million fee for the race.
He said the Turkish government is willing to put in $13.5 million, "So we are close to the sum of about $20 million a year".
Ecclestone said: "It is an issue that we would be happy to discuss with those responsible.
"We would very much like to return to Turkey."