They foresee proposals for a suspension of negotiations on the Comprehensive Air Transport Agreement and a halting of high-level dialogues for the time being.
"We express grave concern over Turkey's announced intention to carry out drilling activities within the exclusive economic zone of Cyprus", EU diplomatic chief Federica Mogherini has said in a statement. The European Council said it was considering "appropriate measures" and would respond "in full solidarity with Cyprus" in light of the second exploration.
The EU had previously claimed that Turkey's activities regarding hydrocarbons in the Eastern Mediterranean are illegal and the union will take some measures on that.
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The EU depends on Turkey to keep a lid on migration from the Middle East to Europe, and for security matters, but the relationship has turned sour over President Tayyip Erdogan's sweeping crackdown on critics following a failed coup in 2016.
"The drilling activities of our ship Yavuz are based on legal and legitimate grounds", the Turkish foreign ministry said in a statement.
Cyprus has argued that while Turkey occupies the northern part of the island, the waters off its coast remain its exclusively, with no connection to Turkey.
The government has instructed police to investigate a reported attack against a Turkish Cypriot bus at Larnaca airport where it was picking up tourists to take to the breakaway north, the government spokesman said on Thursday.
The softer tone was adopted after some European Union countries said they didn't want to tangle with the tough Erdogan, preferring to issue press releases instead.