Zaev said the deal would open the way for the tiny Balkan nation's eventual membership of the European Union and North Atlantic Treaty Organisation, now blocked by Greece's objections to its use of the name Macedonia.
Today (June 12), leaders from both nations said a final agreement may well have been reached. He has been a representative of the United Nations secretary general on the same issue since 1999. Macedonia came to be known as the Former Yugoslav Republic of Macedonia, or FYROM.
Officials in Albania are hailing the agreement between bordering Greece and Macedonia to end their decades-long name dispute as good for the whole Western Balkans region.
A man holds a Greek flag next to a statue of Alexander the Great.
"This is a bad agreement that is in conflict with the majority of the Greek people", he said.
NATO and European Union officials welcomed the breakthrough, which NATO Secretary General Jens Stoltenberg said would help consolidate regional peace and stability.
"We wholeheartedly congratulate Prime Ministers Alexis Tsipras and Zoran Zaev for their determination and leadership in reaching this historic agreement", said EU foreign affairs chief Federica Mogherini and European Commissioner for Enlargement Negotiations Johannes Hahn in a joint statement.
Macedonia has agreed to change its name to Northern Macedonia in order to end a decades-old row with Greece, which among other things prevented the former Yugoslav republic from joining the European Union and North Atlantic Treaty Organisation.
Greek Prime Minister Alexis Tsipras [left] and his Macedonian counterpart Zoran Zaev [right] in Sofia in May. But the new deal will also emphasize the fact that Macedonians to the north speak a Slavic language and don't have a linguistic kinship with the Macedonians of Greece.
He said that the accord would allow a clear distinction between Greece's Macedonia province and the country.
Prime Minister Alexis Tsipras said Tuesday that a deal had been reached to give Macedonia a different name, addressing all of Greece's concerns.
"We have a deal". He did not, however, spell out what the name will be.
In a nutshell, Greek people felt like the name belonged to them because it ties in with the country's heritage and the Macedon Kingdom - the king of which was Alexander the Great, who you'll have heard of.
The dispute has poisoned relations between the two neighbors since Macedonia's independence from Yugoslavia in 1991.
Greek Defence Minister Panos Kammenos, a hardline junior coalition partner in Tsipras's government, earlier dismissed talk of a deal, saying there was "no chance" Zaev could get it approved.
Macedonia had proposed four names to resolve the dispute: Republic of North Macedonia, Republic of Upper Macedonia, Republic of Vardar Macedonia and Republic of Macedonia (Skopje).
Organizers of past rallies in Greece's main cities against a compromise with Macedonia also expressed outrage at the deal, with one accusing Tsipras of "high treason".