Dujarric, in a statement on his website, said: "Both leaders have accepted his invitation. The secretary general looks forward to welcoming the leaders, together with his special adviser on Cyprus, Espen Barth Eide," on June 4.
A UN drive to hold a crunch Cyprus peace conference in Geneva collapsed last week, leaving the future of two years of talks on reunifying the island in limbo.
Eide has since travelled to Athens and Ankara to try to rescue the peace process.
It was agreed that talks on Cyprus could go no further and only an international conference involving the three guarantors of the island's sovereignty -- Britain, Greece and Turkey -- could achieve real progress.
A previous conference involving the guarantor powers in Geneva in January failed to agree on a post-peace security strategy.
The UN-backed talks seek to reunite Cyprus under a federal roof.
Anastasiades and Akinci remain at loggerheads over core issues such as power sharing, territorial adjustments, security arrangements and property rights.
Anastasiades, who heads the island's internationally recognised government, faces re-election in February, a factor complicating the talks process.
The government's drive to explore for offshore oil and gas has also clouded the negotiations, with Ankara calling for it to be halted until a settlement has been reached.
The two sides have been engaged in fragile peace talks since May 2015 that observers see as the best chance in years to reunify the island.
Much of the progress until now has been based on the strong personal rapport between Anastasiades and Akinci, leader of the breakaway Turkish Republic of Northern Cyprus.
But in recent months there has been a negative climate of blame and mistrust.
The eastern Mediterranean island has been divided since 1974 when Turkish troops invaded its northern third in response to an Athens-inspired coup seeking union with Greece.
After a UN reunification blueprint was rejected by Greek Cypriots in a 2004 referendum, Cyprus joined the EU still a divided island, with the breakaway north recognised only by Turkey.