"Regarding Turkey's drilling in the eastern Mediterranean, we have a very clear position - international law must be respected, so progress in EU-Turkey relations is only possible if Ankara stops provocations in the eastern Mediterranean," German Foreign Minister Heiko Maas said during a visit to Athens.
Turkish drilling off Cyprus must stop, he said.
The EU is unhappy at what it says is Turkey's illegal drilling for oil and gas off the coast of Cyprus, as well as Ankara's actions in support of the UN-recognised government in Libya and accusations the Turkish government is eroding rights and democracy at home.
Egypt's parliament on Monday greenlighted a possible deployment of troops beyond its borders, to counter potential military operations in neighbouring Libya, a move which Maas said could "escalate" matters.
There is also concern at the direction Turkey - still nominally a candidate to join the EU - has taken under President Recep Tayyip Erdogan, with the decision to turn Istanbul's iconic Hagia Sophia back into a mosque the latest source of contention.
Hours after the German foreign minister's comments, the Greek government said it had instigated a "protest process" against Turkey over "the announcement of exploration in part of the Greek continental shelf".
"We call on Turkey to stop these illegal activities which violate our sovereign rights and undermines peace and security in the region," the Greek foreign ministry said in a statement.
Read also: Turkey applies for oil exploration licenses in Libya
EU diplomatic chief Josep Borrell, who travelled to Ankara for talks with Turkish ministers last week, has stressed the need to defuse tensions through dialogue.
Maas on Tuesday also insisted on the "importance" of keeping dialogue channels open with Turkey, a "strategically important" country on NATO and migration issues.
Germany, which took over the rotating six-month European Council presidency on July 1, is also placing migration high on its agenda.
"We want a common European answer on the migration issue," the German minister said.
Turkey, which hosts nearly four million Syrian refugees on its soil, accuses the EU of not honouring all its commitments under a controversial 2016 agreement to combat people smuggling in return for financial assistance.
Erdogan announced in February the opening of the border with Greece, causing a surge of tens of thousands of refugees at the frontier.
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