Rival Cypriot leaders returned to the negotiating table Tuesday to press on with a bid to end their country's 42-year-old division, but hopes of a deal hung in the balance.

"We are within reach of an agreement," Cyprus government spokesman Nicos Christodoulides said, cited by local media.

But he added: "We are as close as we are far."

As he arrived at the UN's European headquarters in Geneva for a second day of talks with Turkish Cypriot leader Mustafa Akinci, Cypriot President Nicos Anastasiades described the discussions so far as "constructive".

But he acknowledged that "we have not yet reached" agreement on sufficient issues to seal a deal.

On Tuesday, the two leaders were poring over the issue of how a reunited Cyprus should be governed.

"You can understand that we have difficult and sensitive issues. There is a will from both sides to have progress to reach agreements or understanding," Cyprus government spokesman Nikos Christodoulides was quoted as saying by the Cyprus Weekly Newspaper.

In the divided Cypriot capital of Nicosia, hundreds of Greek and Turkish Cypriots held a peace rally between checkpoints to urge reunification.

"Nico-Mustafa, come back with a solution," read a poster.

The Beatles' hit "Come Together" played on the loudspeaker before the rally, which was called by dozens of associations, unions and parties from both sides of the island.

Akinci and Anastasiades have been negotiating for more than 18 months, in what many commentators say is a historic opportunity for reunification.

But deep divisions remain on core issues such as property, territorial adjustments and security.

Cyprus has been divided since Turkish troops invaded in July 1974 in response to an Athens-inspired coup seeking union with Greece.

This is the third time the Cypriot leaders have met in Switzerland since November, but the two previous rounds were inconclusive.

And the two sides are still facing a range of thorny issues that have blocked progress for decades, including how to redraw boundaries and ensure security on the island.

- Maps on the table? -

The three days of talks are set to wrap up Wednesday with the parties presenting maps of their proposals for the internal boundaries of a future bi-zonal federation on the eastern Mediterranean island.

If that goes to plan, they will be joined from Thursday for an international conference chaired by the UN's new secretary general, Antonio Guterres, and attended by representatives of the island's three guarantor powers -- former colonial ruler Britain, Greece and Turkey.

"It's going to be the first time in the history of the Cyprus problem that we are going to have such an important conference," said Cyprus government spokesman Christodoulides.

But he confirmed that Greek Prime Minister Alexis Tsipras is not currently planning to attend, and Cypriot media said the participation of Turkish President Recep Tayyip Erdogan was also highly uncertain without significant progress.

London meanwhile said that British Foreign Secretary Boris Johnson, not Prime Minister Theresa May, is due to attend.

- 'Moment of truth' -

"We are now really at the moment of truth," Espen Barth Eide, UN envoy for Cyprus, said Monday, while warning that some of the "most complicated or most emotional issues" remained to be agreed upon.

On Tuesday, the two delegations were discussing the island's relations with the European Union as well as a future system of government for a prospective federal state and the economy, the UN said.

While Cyprus has been an EU member since 2004, Anastasiades's internationally recognised government exercises no control over the northern Turkish-ruled part of the island, and EU legislation is suspended there until a settlement is reached.

The EU therefore has a vested interest in seeing the Cyprus conflict resolved, and European Commission chief Jean-Claude Juncker is planning to attend Thursday's conference as an observer.

On Monday, the two sides tried to thrash out an agreement on the difficult issue of property.

But they appear to remain far apart on how many Greek Cypriots should be able to return to homes they fled in 1974, with Akinci determined to minimise the number of Turkish Cypriots who would be displaced for a second time.

There are also still significant differences over security, with Anastasiades wanting Turkish troops to leave the island but Akinci determined to keep a military presence.