Eni holds an 80 percent stake in three reserve areas in the deep waters in the Levantine basin off the coast of Cyprus. It is working alongside Korean natural gas company KOGAS.
Eni Chief Executive Officer Paulo Scaroni met in Cyprus with Anastasiades to discuss future development in the country "in the case of a discovery."
The Italian energy company said Thursday it would meet again with Cypriot officials to discuss the progress of developments in the Levantine basin.
Eni made its debut in Cyprus in January with the three deep-water blocks, which cover about 4,800 square miles.
Offshore reserves in the region may be a source of contention given regional rows between the Cypriot and Turkish governments. The Cypriots had discussed coordinating with their Lebanese counterparts on offshore reserves.
Cyprus is considered frontier energy territory. Its offshore area could hold nearly 12 trillion cubic feet of natural gas reserves.