Erdogan said on Sunday: “We continue and will continue to search in those areas, Cypriot waters.”
However, an EU draft statement will say that Turkey “continues to move further away from the European Union” and calls on Ankara to stop “illegal” drilling in the region.
It will then call on EU leaders “to respond appropriately and in full solidarity with Cyprus”.
Sources said that Cyprus was willing to block various resolutions and texts if the EU fails to take a harder line on Turkish behaviour.
Brussels may formally end talks on upgrading its customs union with Turkey – which ensures more seamless trade between Ankara and the EU bloc – and visa-free travel.
The charge has been led by Greek Prime Minister Alexis Tsipras, who came to the defence of his neighbouring country.
He said: “We have agreed to prepare the ground in the coming week that the summit take the relevant decisions, even sanctions against Turkey, if it is verified that there has been a drill in the Cypriot EEZ.”
His defence minister added today: “It is not easy, but we will make this effort to convince Europe to adopt this direction.”
He was supported by France, Greece, Italy, Portugal, Spain and Malta, who all released a joint statement following a summit of EU Mediterranean countries.
The Turkish leader lambasted French President Emmanuel Macron on Saturday after he waded into the debate to criticise Ankara.
He added: “Those who have the right to speak about the eastern Mediterranean can speak. When did France have the right to speak on the eastern Mediterranean? Do they have a coast in the eastern Mediterranean?
“We are one of the guarantor countries in Cyprus. Greece and the United Kingdom are also the guarantor countries. What is France doing there?
“Are you [Macron] making such statements for Total?
“It means you have left the presidency and have now begun work as a lawyer.”
A Turkish drilling ship has reportedly been operating in the EEZ since early May, but the EU has been relatively passive on the issue until now.
Cyprus itself is a flashpoint of conflict between Turkey and Greece, with the country being split in 1974 between a Greek Cypriot south – which is part of the EU – and a Turkish Cypriot north.