He told reporters "we expect tomorrow's meeting to draw up a road map pointing us to the way to the long-expected final resolution" of the conflict.
"Both sides want Ban to act," he said, adding that the UN secretary general had "reaffirmed his commitment" to the negotiation process.
Akinci reiterated that the goal was to secure a deal by the end of the year, as a new UN chief and US president would be taking office in January and would need time to get to grips with the Cyprus issue.
Cyprus has been divided since 1974, when Turkey invaded the northern part of the island following an Athens-inspired coup by Greek Cypriots coup seeking union with Greece.
The division is one of the world's most intractable diplomatic problems.
Much-delayed UN-brokered talks were relaunched in May 2015, and they are seen as the best chance yet to end four decades of division.
Sunday's talks will gather Ban, Akinci and Greek Cypriot President Nicos Anastasiades.
Ban's envoy, Espen Barth Eide, announcing the meeting on September 14, said the leaders had made "significant progress" on outstanding issues, including governance and power-sharing, but remained divided on the thorny issues of property, security and territorial adjustments.
It has always been agreed that the Turkish Cypriots will cede territory controlled by their breakaway state in the north to the Greek Cypriot component of a future federation.
But which areas are to be handed over is a sensitive issue, as they will determine the number of Turkish Cypriots who will be displaced from their homes.
Other major hurdles are how many people will be allowed to return to their former homes and how many will be entitled to financial compensation.
The costs of compensation are estimated to run to billions of euros (dollars). Eide said the financial aspects of a settlement would also be discussed in New York.
Any agreement the two leaders reach will have to be put to simultaneous referendums on either side of the island.
A previous peace deal brokered by then-UN chief Kofi Annan in 2004 was backed by a significant majority of Turkish Cypriot voters but overwhelmingly rejected by their Greek Cypriot counterparts.
The outcome meant that Cyprus joined the European Union later that year still a divided island. The northern one-third is governed by the self-proclaimed Turkish Republic of Northern Cyprus, which is recognised only by Turkey.
