The court had been due to make a decision Monday but judges postponed the case after one judge suffered an accident.
The soldiers, who deny involvement in the coup bid, fled Turkey on the night of July 15 in a Blackhawk helicopter and landed in Greece.
They claimed asylum on the grounds that their lives were at risk in Turkey. Their case is being heard by the Supreme Court after previous rulings to extradite three and not extradite the other five were contested.
If the court decides they should be extradited their asylum applications will be void. Thursday’s decision will be final and may only be appealed at the European Court of Justice.