France also criticised its NATO ally over a drone strike this week in northern Iraq, while Turkish President Recep Tayyip Erdogan accused French counterpart Emmanuel Macron of seeking to "bring back a colonialist structure" as he visited Lebanon.

The United States called for a lowering of tensions between France and Turkey, describing them as "incredibly important NATO allies".

But the two countries have been at loggerheads for months over Turkey's military intervention in Libya to support the UN-recognised Tripoli government, with France saying Turkey was bringing the NATO alliance's credibility into question.

France has also been critical of what it says are Turkey's violations of the sovereignty of Greece and Cyprus.

Ankara is at odds with Greece and the European Union over maritime rights in the Eastern Mediterranean over gas reserves.

The French defence ministry said two jets would arrive Thursday on the island of Crete for a stay of "several days", and that French military vessels took part in joint exercises with the Greek navy overnight.

The moves were designed "to affirm France's commitment to free movement, to the security of maritime navigation in the Mediterranean and respect for international law", said a ministry statement.

Last month, Greece announced it had deployed ships in the Aegean in "heightened readiness" after Turkey announced plans for energy exploration near a Greek island in an area it claims is within Turkey's continental shelf.

Then on Monday, Ankara dispatched the research ship Oruc Reis, accompanied by naval vessels, off the Greek island of Kastellorizo.

Greece deployed warships to monitor the Turkish vessel and Foreign Minister Nikos Dendias urged Ankara to "immediately" remove it from Greek waters.

The US defence department called for Turkey and France to work in harmony.

"We would like to see them continue to work together and find solutions that do not involve the necessity of having warships or have aircraft deployed in a less than cooperative environment," Pentagon spokesman Jonathan Hoffman said.

- 'Risk' of an accident -

Macron told Greek Prime Minister Kyriakos Mitsotakis on Wednesday that Turkey's "unilateral decisions" on gas exploration "must be stopped" to allow peaceful discussions within NATO, according to the French presidency.

France, Turkey and Greece are all members of the NATO military alliance.

In his talks with Mitsotakis, Macron reiterated French and EU solidarity "for any member state whose sovereignty is challenged".

Mitsotakis said on Wednesday he hoped sense would prevail and dialogue resume, warning: "The risk of an accident lurks when so many military assets are gathered in such a contained area."

Erdogan spoke to German Chancellor Angela Merkel and European Council President Charles Michel on Thursday about defusing the row with Greece.

Erdogan "stressed that he supported solving problems in the Eastern Mediterranean within the framework of international law and the principles of dialogue and equality", the presidency said.

But he criticised Macron for "putting on a show" with his visit to Beirut.

"Macron's concern is to bring back a colonialist structure," the Turkish leader said, referring to the years Lebanon had spent under French mandate after the collapse of the Ottoman Empire.

On Thursday, the foreign ministry in Paris criticised a drone strike in northern Iraq that Baghdad labelled a "blatant Turkish drone attack" in its autonomous Kurdish region.

Iraq said the strike killed two of its high-ranking officers.

"France is deeply attached to full respect for Iraqi sovereignty," said the French ministry. "France condemns any violation of this sovereignty."

burs-mlr/sjw/jxb