Diplomatic efforts to reunite the island have failed after it was split into northern and southern territories in 1974 after Turkey intervened militarily following a Greek-inspired coup designed to annex Cyprus to Greece.
The head of the United Nations flew into Switzerland on Thursday to press Greek and Turkish Cypriots to seal a deal reuniting their east Mediterranean island. The US vice president urged them to "seize this historic opportunity".
Diplomatic efforts to reunite Cyprus have failed since the island was riven in a 1974 Turkish army intervention triggered by a coup by Greek Cypriots seeking union with Greece.
But factors such as the discovery of gas in the area could increase pressure – domestic and international – for a deal.
UN Secretary General Antonio Guterres flew into the Swiss resort of Crans-Montana, where Greek Cypriot leader Nicos Anastasiades and Turkish Cypriot leader Mustafa Akinci have been meeting.
He was due to leave for the G20 summit on Friday morning, opening only a narrow window for what might be a "final push".
"The day is almost over. Let's see what the night will bring," Akinci told reporters as the leaders moved to a working dinner, which is expected to be followed by more talks.
"Negotiations are continuing, and I want to say one thing; cool-headedness and patience is required," Anastasiades said.
Greek Foreign Minister Nikos Kotzias said the Greek side had done all it could to find a solution. "It's a very difficult exercise," he said.
Earlier in the day, US Vice President Mike Pence rang the rival Cypriot leaders to underscore US support, the White House said in a statement.
"The Vice President urged President Anastasiades and Mr Akinci to seize this historic opportunity to reunify the island and expressed his confidence in both leaders' ability to secure a settlement that would reunify Cyprus as a bizonal, bicommunal federation," the statement said.
Recent major gas discoveries in the Eastern Mediterranean have refocused attempts to end the conflict, partly because the basin could hold sufficient to wean Europe off reliance on Russian gas, but also because of overlapping territorial claims.
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 war.
One diplomat at the talks said that while there were signs Turkey might be willing to reconsider its intervention rights, it did not appear to waver on the residual presence of Turkish troops on the island after a settlement, a sticking point for the Greek side which wants assurances of their full withdrawal.
A "sunset" clause for a full withdrawal of Turkish troops or a further review of whether they are required in Cyprus after a given period of time would be the key bargaining issues, this diplomat said.
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 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.