The European Union’s executive body on Monday called for dialogue with Turkey and demanded Ankara stop stoking tensions in the eastern Mediterranean.
The EU has already moved towards sanctions if dialogue did not help calm the dispute, a spokesman for the European Commission told a briefing on Monday.
Foreign ministers from each of the EU’s 27 member states last week discussed the rising tensions between Greece and Cyprus and Turkey.
A senior EU diplomat, speaking on condition of anonymity, on Friday said: “As well as sticks (sanctions), there will be carrots too to get Ankara to engage seriously in dialogue. These carrots could be progress on a new customs union and more money for the refugee programme.”
In the latest escalation of the row over gas exploration in the region, Turkey on Monday accused Greece of “piracy” and attempting to militarise an island at the centre of the dispute.
Greece claims that the waters around the island – known as Kastellorizo in Greece and Meis in Turkey – are under Greek sovereignty and opposes any Turkish exploration nearby. The island sits two kilometres off the Turkish coast.
Images published in the media last week showed Greek soldiers arriving on the Island.
"Greece's attempt to arm the island of Meis is an example of a new piracy," said Omer Celik, spokesman for Turkey's ruling Justice and Development Party.
"Greece will learn we will have a strong answer to every step it takes," he added.
The Turkish foreign ministry on Sunday said the island was "under a demilitarised status established with the 1947 Paris Peace Treaty".
"Such provocative actions will prove useless for Greece," the Turkish ministry added.
Athens has issued no official comment on the two neighbours' latest dispute.
War games roll on as tensions between Ankara and Paris rise
Tensions between Greece and Turkey - who are both Nato members - have been rising since Ankara encouraged migrants to flee to Europe by opening its western border in February.
Turkish President Recep Tayyip Erdogan on Sunday marked the 98th anniversary of the War of Independence with Greece with inflammatory comments framing the dispute as a "struggle for independence".
Both Greece and Turkey have staged large-scale military exercises in the eastern Mediterranean in recent weeks.
Greece and Cyprus were joined by France, Italy and the United Arab Emirates in carrying out naval and aerial war games in the area.
Turkey began its own manoeuvres off its south coast on Saturday, which are due to last until September 11.
Tensions are also growing between Turkey and France, another Nato ally, which has thrown its weight behind Greece.
French President Emmanuel Macron said on Friday that Paris had adopted a "red-line policy" for Turkey in the eastern Mediterranean, though he also called for dialogue.
"When we talk about sovereignty in the Mediterranean, I must be consistent in actions and words... the Turks only consider and respect that," Mr Macron said.
But Ankara said Turkey would not be pushed around by the French leader.
"Those, who think to have drawn red lines against the righteous cause of Turkey in the eastern Mediterranean, will only face Turkey's firm stance," Turkish foreign ministry spokesman Hami Aksoy said in a statement late on Sunday.
He called for "those, who are in delusion of grandeur, to face reality," adding that "the era of defining imperialist conceptions by drawing lines on maps is long gone."