Nicos Anastasiades, a Greek Cypriot, says he and breakaway Turkish Cypriot leader Mustafa Akinci have made significant progress on numerous issues making an envisioned federation workable.
Those issues include ensuring the country's economic viability and the right of all citizens to live and work wherever they chose.
But he says it'll take the "resolute contribution" of Turkey and the Turkish Cypriot leadership to reach agreement on core issues such as security and how much territory either side will administer under an envisioned federal state.
Officials say sufficient progress on this key issue would pave the way for a final summit to hammer out a comprehensive deal encompassing security matters.
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UN Secretary-General Ban Ki-moon will attend Monday's opening session.
"This is a critical juncture in the talks and he welcomes very much the fact that the two leaders have jointly expressed their hope that this meeting will pave the way for the last phase of the talks," UN spokesman Stephane Dujarric said in New York.
A Turkish invasion in 1974 following a coup aiming at union with Greece split the eastern Mediterranean island into a breakaway Turkish-speaking north and an internationally recognized Greek-speaking south.