Alexander Downer, the UN envoy for Cyprus who facilites negotiations, said there was "no doubt that a Cyprus solution is doable".
Turkish troops invaded Cyprus's north in 1974 after a brief Greek Cypriot coup engineered by Greece's former military government.
Currently, the two populations each govern a portion of the island. The goal of peace talks is to re-link the island as a federation.
"To achieve an agreement will require a great deal of political strength, courage and will by both sides," Downer, reading out Ban's statement, said.
A deal "is not beyond their grasp, it is within their reach".
Jacky Rowland, Al Jazeera's correspondent reporting from the meeting, was less optimistic.
"In the past, the Greek Cypriot leader held 71 meetings with the Turkish Cypriot leader and they were unable to clinch a deal," Rowland said at the beginning of the latest talks.
A deal in Cyprus is crucial for Turkish efforts to join the European Union.
"The Greek Cypriot side is already a member of the EU and current members can veto new members," our correspondent said.
Greek Cypriots say Turkey cannot join the EU without a re-unification deal on the island.
Presently, Turkey is the only country to recognise the Turkish enclave on the island's north as a separate state.
There are still around 30,000 Turkish troops on the island.
The issue of property will be crucial in negotiations between the two sides.
"Before the Turkish invasion 90 per cent of property belonged to Greek Cypriots," Rowland said.
"Should homes go back to their original owners? Or do people who have been living in them have a claim?"
The election of Eroglu, unseating Mehmet Ali Talat as Turkish Cypriot leader who was seen as a moderate, unnerved some Greek Cypriots.
Eroglu, a nationalist Turkish Cypriot leader, advocates greater independence for his community. Greek Cypriots want a single state with two self-administering areas.