Davutoğlu, currently in Athens to attend meeting of Organization for Security and Cooperation in Europe (OSCE) Ministers Council, met with Papandreou for 45 minutes.
Davutoğlu gave a news conference after his meeting and described the meeting as "very fruitful".
"The parties shared the vision pertaining to the future of the two countries. I see a very strong ground to boost Turkish-Greek relations," Davutoğlu said.
Davutoğlu said high level political dialogue between Turkey and Greece would continue increasingly, economic relations would be strengthened and measures would be in question to foster cultural relations.
Foreign Minister Davutoğlu said Turkey developed cooperation council with other neighboring countries noting that it would be in the interest of the both parties in case a similar cooperation mechanism could be established between Turkey and Greece.
"I have explained this to Papandreou. Greek party said it would assess the matter," he said.
Davutoğlu said foreign ministries of Turkey and Greece would carry out negotiations within this framework.
Turkish foreign minister said, "we have exchanged views on problems in the Aegean, exploratory talks, Cyprus question, mutual contribution to solution of Cyprus question, as well as roles to be assumed as countries having observer status aiming to find a fair and lasting solution in Cyprus. It is important to relay encouraging messages to the parties during negotiation process."
Davutoğlu said Turkey and Greece had very important roles to turn east Mediterranean into an area of peace, security and prosperity.
Davutoğlu said the parties "sincerely" expressed what should be done on minorities in Turkey and Greece and cultural relations.
Turkish foreign minister thanked to Greece for the support it extended to Turkey's EU bid.
Referring to the climate summit that would take place soon, Davutoğlu said, "Turkey and Greece share the same fate as they share the same geography. We agreed that more negotiations should take place in international organizations regarding the matters affecting common future."
Davutoğlu will proceed to Brussels on Wednesday evening to attend NATO foreign ministers' meeting.