On Monday, diplomats will meet to negotiate an agreement after Turkey failed to meet a treaty obligation to open its ports and airports to traffic from Cyprus.
Most favour partial suspension of entry talks rather than a freeze.
Ahead of the talks, Dora Bakoyanni, the Greek foreign minister, said: "Our central aim is ... a common European position which will include a clear and unwavering message on the future of Turkey's negotiations with the EU."
Should Monday's talks be inconclusive, the decision will be made by EU leaders at a summit in Brussels later this week.
Powers within the union are split over Turkey's entry.
The European Commission has proposed suspending eight of 35 negotiating areas, or "chapters", but Cyprus, Greece and Austria have said that would not be enough and called for a new deadline for Turkey to normalise trade relations with Cyprus.
Supporters of Turkey have said the EU executive's proposal is too harsh.
Margaret Beckett, the British foreign secretary, wrote in the Turkish Sabah newspaper: "We must not make a mistake here because if we push Turkey away, in the end it will lead to a much poorer, weaker and less secure EU."
Turkey has said it would lift the embargo on Cyprus-registered ships and open its airspace to Cypriot planes only if the EU made good on its pledges to ease the economic isolation of the Turkish Cypriot state in northern Cyprus.
Last week, Turkey offered to open one major port and possibly an airport to traffic from Cyprus and said it would do more if the EU allowed direct trade with the Turkish Cypriots.
EU president Finland said that was not enough and Cyprus rejected the offer.
The EU admitted Cyprus, represented only by the Greek Cypriot government, to the union in 2004. Monday's talks will also focus on the Balkan states, including EU relations with Croatia and Serbia.
