The statement, by Foreign Minister George Papandreou, came a day after the European Commission said Turkey was not yet ready, and in the wake of overnight pressure from the US.
Washington urged the EU to hold membership talks with Turkey as soon as possible, describing the country as a vital ally in Europe.
Turkey itself had already expressed its anger at the European Commission's recommendations to member states to admit 10 new members in 2004, but not to set a date for talks with Ankara.
"I believe a positive message to Turkey must be sent from [the EU summit in] Copenhagen," Mr Papandreou told Greek radio.
Asked if this message could even be a specific date for the start of negotiations, he said: "Why not, we can say even that."
Click here to see a map of EU applicants
Relations between Greece and fellow Nato alliance member Turkey have improved in recent years, easing their traditional enmity.
BBC Athens correspondent Panos Polyzoidis says Greece would rather deal with difficult issues such as the division of Cyprus with Turkey inside the EU than out.
The Turkish President, Ahmet Necdet Sezer, called on the EU on Thursday to show "common sense" and set a date for negotiations when the member countries meet in Copenhagen.
He said that allowing predominantly Muslim Turkey into the European Union would help Europe tackle cultural differences.
The Turkish Government warned it might revoke agreements with the EU if its progress towards membership was delayed.
Turkey made extensive reforms to its legal system over the summer, abolishing the death penalty and lifting restrictions on the use of the Kurdish language.
But the EU report said Turkey had failed to meet the criteria necessary to join the current round of EU expansion, noting in particular its human rights record.
~RS~a~RS~International~RS~q~RS~~RS~z~RS~08~RS~)