A spokesperson for Eni told the Cyprus News Agency that Turkish military ships had interfered with drilling plans in the block using the Saipem 12000 drillship.
It said Ankara had in fact contacted Athens regarding the "dangerous maneuvers" by the Greek coast guard, and informed them that Turkey "would not tolerate continuing hostile behavior by the Greek armed forces".
"The EU also stresses the need to respect the sovereignty of member states over their territorial sea and airspace".
Supermajors Eni and Total have made a promising gas discovery offshore Cyprus last week that confirms that the Zohr-like play, where Eni found the biggest gas deposit in the Mediterranean offshore Egypt, extends into the Cyprus Exclusive Economic Zone, the Italian group said.
Former vice-admiral of the Turkish navy, Cem Gurdeniz, said in an interview to Turkish newspaper Milliyet on Sunday that the harassment of Saipem 12000 is a serious demonstration of the power of the Turkish fleet.
Turkey and Cyprus have faced-off regarding natural resources in the Mediterranean in the past, especially because of Ankara's ongoing argument that the northern portion of the island belongs to Turkey.
"From our side, our actions reflect the necessity of avoiding anything which could escalate (the situation), without of course overlooking the violation of global law perpetrated by Turkey", Anastasiades told journalists in Nicosia.
He said that his government would take the "necessary steps", but added that he wanted to "avoid anything which could escalate the crisis".
Turkey rejects what it says is a "unilateral" search for resources by the Cypriot government, saying that it forgets the rights of the retracted Turkish Cypriots to the island's natural resources, threatening security and stability in the region.
The Turkish government said the blame for any situation that could arise from the offshore stalemate rests with Cyprus. "Such a thing happened last night again".
Turkey's Europe minister, Omer Celik, will meet with European Union foreign affairs ministers and European Union high representative Federica Mogherini in Sofia later this week.