Greek Cypriots recently indicated they would go ahead with plans for gas exploration, despite opposition from Turkey .
One diplomatic source said Turkey might be willing to rethink its role on guarantor rights. Turkey, Greece and Britain are guarantor powers of Cyprus in an independence treaty which granted the former colony independence in 1960.
That treaty gives them intervention rights to restore constitutional order.
Turkey has about 30,000 troops in northern Cyprus, a holdover from the 1974 invasion.
Greek Cypriots had previously objected to Turkish Cypriot demands for a rotating presidency, but another source said they had indicated readiness to discuss such an arrangement under conditions.
Any agreement would almost certainly have to go to separate referendums in the Greek and Turkish parts of the island, which each have their deeply-rooted sensitivities born of past communal conflicts. Negotiators must agree what powers would be held by two probably largely autonomous zones and which would be ceded to any central administration.
It was the second time in a week negotiators had drafted Guterres in to lend his weight to the talks.
“The problem has always been who goes first with concessions that will allow the others to follow,” a diplomat at the talks said.
“It is now obvious that only the Secretary-General has the authority and stature to press the various parties and perhaps find a way to break this deadlock.”