Greece and France have both demanded tough sanctions against Ankara, which could be agreed at an EU summit in October.
It is recalled France sent fighter jets and vessels to the Eastern Mediterranean region in support of Greece, which angered Turkey.
According to government officials, French President Emmanuel Macron and Turkish President Recep Tayyip Erdogan spoke over the phone late on Tuesday.
French officials said Macron told Erdogan to avoid taking any action likely to raise tension and to "engage without ambiguity in the construction of an area of peace and cooperation in the Mediterranean".
Turkish officials said Erdogan told his counterpart he expected "common sense and a constructive attitude" from France, claiming that he supports dialogue and cooperation to solve problems.
Two months ago, Macron expressed his solidarity with Cyprus and Greece in a message posted on Facebook written in Greek.
https://www.facebook.com/EmmanuelMacron/posts/2842476725984895
Meanwhile, Turkey and Greece have agreed to talks over disputed waters in the eastern Mediterranean.
Both sides said they were "ready to start exploratory talks" as a three-way videoconference between Turkish President Recep Tayyip Erdogan, German Chancellor Angela Merkel and European Council chief Charles Michel took place.
The Greek Foreign Ministry also confirmed the talks, saying they would be held soon in Istanbul, without providing any precise date.
