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The Finnish government - the current president of the EU - has offered to host talks between some key players.
The European Commission is to produce a report next week on Turkey's progress in its membership negotiations.
There are already some voices in the EU calling for the talks, which opened in October last year, to be suspended.
In return for the start of the talks, Turkey was supposed to open its ports and airports to all EU traffic, including that from the divided island of Cyprus. But the Cypriots are still waiting for access.
One EU diplomat described the current draft of the Commission's report on Turkey as highly critical.
There are hopes that Turkey may agree to open its ports and airports if the EU's economic isolation of Turkish-controlled northern Cyprus is lessened somewhat.
Even so, the Commission's report card is also likely to bash Turkey for its record on human rights and free speech.
This process was supposed to be, as much as anything, about the symbolism of Europe and Asia, the West and Islam, working together.
That is why diplomats are describing the current situation as very high-risk. BBC