French President Emmanuel Macron has urged Turkey to halt oil and gas exploration in disputed waters in the area. A Turkish survey ship began such a mission on Monday, angering Greece.
Mr Macron told Greek Prime Minister Kyriakos Mitsotakis that the French military would monitor the situation.
The area is rich in untapped energy.
Turkish President Recep Tayyip Erdogan said the only solution in the Mediterranean was dialogue and that his country was not chasing adventure.
"If we act with common sense and reason, we can find a win-win solution that meets everyone's interests," he said.
There are also tensions around Cyprus over rival exploration rights. The Republic of Cyprus and Greece do not accept any such rights for Turkish-controlled northern Cyprus in the region.
France is also at odds with Turkey over the crisis in Libya. Turkey has sent military support to the UN-recognised government in Tripoli, while France, Russia, Egypt and the United Arab Emirates support the forces of Gen Khalifa Haftar. Russia and the UAE are Gen Haftar's main arms suppliers.
France already has a helicopter-carrier, Tonnerre, heading to Beirut with aid to ease the city's suffering after the devastating 4 August port explosion.
The French frigate La Fayette has been on exercises with the Greek navy and is staying in the area. The Rafale jets were in Cyprus for an exercise and are now relocating to Souda, on the Greek island of Crete.
Mr Macron tweeted: "I have decided to strengthen the French military presence temporarily in the Mediterranean, in co-operation with Greece and other European partners.
"The eastern Mediterranean situation is worrying. Turkey's unilateral decisions concerning oil exploration are provoking tensions. Those tensions must end, to enable calm dialogue between countries which are neighbours and allies in Nato."