The United Nations in New York said Wednesday that UN envoy Matthew Nimetz would travel to Greece and Macedonia next week to push for a solution to the 27-year-old dispute.
Greece's objections to its neighbour becoming independent in 1991 as Macedonia -- the name the former Yugoslav province has used since the 1940s -- have hampered the tiny nation's bid to join the European Union and NATO.
Athens argues that the name Macedonia suggests that Skopje has territorial claims to Greece's historic northern region of the same name going back to when Alexander the Great ruled in the fourth century BC.
A Macedonian diplomat said the airport move could mark a major step in mending fences over the name issue.
Since the breakup of the former Yugoslavia, Greece managed to ensure that newly independent Macedonia gained entry to the United Nations in 1993 only under the provisional name the Former Yugoslav Republic of Macedonia.
Athens also used its vote to prevent Macedonia from entering both the European Union and NATO -- inducting new members requires a unanimous vote in both bodies.
Minutes before Zaev made his announcement, Tsipras said his counterpart was about to "take important initiatives" likely to thaw relations so long hamstrung between the two nations.
Citizens discuss Macedonia name row
Earlier, the Greek government had said it hoped that the meeting in Davos between Tsipras and Zaev, brokered by the UN, would "mark real progress" in setting relations on a much more positive footing.
The leftist Tsipras said Athens was out to take steps to bolster that process including the long-mooted opening of a second border crossing.
Greek media reported that UN mediator Matthew Nimetz is due in Athens on Tuesday and would go on to Skopje the following day. Delegates from both nations first met in New York last week.
Compromise country names being touted by negotiators include "North Macedonia" and "New Macedonia".
Tsipras and Zaev's efforts to resolve the issue follow a demonstration Sunday at Thessaloniki, the capital of Macedonia province in northern Greece, against the use of Macedonia by Greece's neighbour to the north.
The protest drew some 90,000 people according to police but 400,000 according to organisers.
The Greek government however said the protest reflected "ultra-nationalist fanaticism" not shared by most Greeks.