Enel has expressed interest in three onshore areas in the western part of the country, Greece's energy ministry said, adding it would duly decide on the Italian company's application.
If permission is granted, Athens will give other oil firms at least 90 days time to submit rival offers for the same areas.
Back in 2001, oil companies including Hungary's MOL and Greece's Hellenic Petroleum searched in vain for oil in two of the blocs Enel is currently considering.
"Enel Group's decision ... is a vote of confidence in our great national effort to exploit our mineral wealth," Energy Minister Yannis Maniatis said in a statement.