Turkey has already opened eight boreholes in the region, Donmez said, adding that while there were signs of natural gas, there had been no economically significant discovery.
"Our experts are checking data after each drill with the seismic data obtained previously. We could have several more drills close to the ones (boreholes) where we see signs of gas," he told broadcaster A Haber.
"Time will tell, but we are hopeful. We evaluate that there is a potential," he said.
Tensions flared between Turkey and EU members Greece and Cyprus over energy resources and jurisdiction in the eastern Mediterranean in August, when both Turkish and Greek navy frigates escorted exploration vessels.
EU leaders in March made good on a 2016 promise to deepen trade ties with Turkey, but also warned Ankara to expect sanctions if it restarts energy exploration in the eastern Mediterranean.
Delivery from a natural gas reserve in the Black Sea, Turkey's largest historical discovery, is expected to begin in 2023.
If the gas can be commercially extracted, the discovery could transform Turkey's dependence on Russia, Iran and Azerbaijan for energy imports.