President Emmanuel Macron voiced concern over "unilateral" exploration by Turkey in a call with Greek Prime Minister Kyriakos Mitsotakis.
The standoff deepened this week with the arrival of Turkish seismic research vessel Oruc Reis in the disputed Mediterranean region, accompanied by warships.
France will temporarily strengthen its military presence, Macron said, to "monitor the situation in the region and mark its determination to uphold international law."
Turkey and Greece are on the brink of competing claims to natural gas reserves, brought into sharp focus by EU member Cyprus's attempts to exploit gas in the eastern Mediterranean over strong objections from Turkey.
Last month, Macron called for EU sanctions against Turkey for what he described as "violations" of Greek and Cypriot sovereignty over their territorial waters.
Maritime Business World