On Tuesday, they sent warships into the waters in a show of force.
Turkish ship Oruc Reis is conducting research – and Greece says it sits above its continental shelf.
Ankara is contesting Athens’ exclusive rights in the region.
The two countries have announced military exercises in a large area between the Greek island of Crete and Cyprus.
But they also signaled a desire for dialogue as Germany tries to mediate between NATO allies.
So could this dispute get worse? And what would it take to defuse it?
Presenter: Adrian Finighan
George Tzogopoulos – Strategic Business Analyst and Lecturer at the Democritus University of Thrace
Wolf Piccoli – Director General of the Political Risk Branch of Global Advisor Teneo
Yusuf Alabarda – Retired Turkish Colonel and Security Analyst at Turkish Defense University
Source: Al Jazeera News