European leaders on Thursday are ready to ask officials to "prepare additional listings" of its Turkey sanctions list.
"Turkish unilateral and provocative activities in the Eastern Mediterranean are still taking place, including in Cyprus's exclusive economic zone [which extends 200 nautical miles from the coast]," according to a draft document seen by reporters in Brussels.
Officials will be urged to back the additional measures on unnamed individuals "in view of Turkey's unauthorised drilling activities in the eastern Mediterranean".
Wider economic sanctions that were threatened in October are unlikely to be introduced, with the EU seeking to co-ordinate with US President-elect Joe Biden on any further action.
The EU supports Greece and Cyprus in their dispute with Turkey over exploration rights in the region. France, Greece, and Cyprus are pushing for strong action against Turkey but have encountered opposition from Germany, Italy and Poland.
"During October's EU summit all leaders decided that if Turkey continues its delinquent behaviour, there will be consequences and we jointly set that the date — when decisions will be taken — will be this EU summit, December's EU summit," Greek Prime Minister Kyriakos Mitsotakis said.
"It will be now seen whether we as Europe are credible on what we among ourselves have agreed," adding a Latin legal phrase. "Pacta sunt servanda (Agreements must be kept)."
Oruc Reis, the Turkish ship at the heart of the tensions, was withdrawn on November 30 in a possible sign that Turkey hoped to resolve the issue, although there has been scepticism about this in Greece.
The issues with Turkey were discussed at the EU Foreign Affairs Council earlier this week. Arriving for the European Council on Thursday, EU foreign affairs chief Jose Borrell said "we cannot provide a positive assessment".
"The behaviour of Turkey has not changed fundamentally. We can even say that, from some points of view, things have been worsening. The assessment is not a positive one and we have to deal with it," he added.
The draft resolution from the European Council – which could still be changed – says that it "trusts that this will be sustained so as to allow for the early resumption of direct exploratory talks between Greece and Turkey”.
It also underlines that the EU is willing for a positive relationship with Turkey if Ankara is also co-operative.
Turkish President Recep Tayyip Erdogan on Wednesday accused the EU of acting dishonestly towards Turkey and he brushed off the threat of sanctions.
“Any decision to impose sanctions against Turkey won’t be of great concern to Turkey,” Mr Erdogan said.
At a summit in October, European leaders warned Turkey to withdraw its energy research ships or face punitive measures.
Two officials are currently on the list, including the Vice-President of the Turkish Petroleum Corporation and the deputy director of its Exploration Department. More people or organisations would be added to the list, the diplomats confirmed, in the weeks ahead.