Tsipras and Zaev spoke on the phone yesterday for an hour, and said they would continue their conversation today.
"We don't agree and we won't vote for any deal including the name 'Macedonia, '" Panos Kammenos, Greek Defense Minister and head of the rightist Independent Greeks party, said.
Greece had long demanded that Macedonia change or modify its name to avoid any claim to the territory and ancient heritage of Greece's northern region of Macedonia - birthplace of ancient warrior king Alexander the Great.
The dispute between the two neighbours had been an obstacle to Macedonia joining either the North Atlantic Treaty Organisation military alliance or the European Union.
"We have been solving a two-and-a-half decade dispute ... that has been drowning the country", he said, adding that the deal "will strengthen the Macedonian identity".
The country was also accepted to the United Nations in 1993 as the Former Yugoslav Republic of Macedonia (FYROM).
In Skopje, meanwhile, Macedonian President Gjorge Ivanov said earlier in the day that he remained opposed to writing the new name into the constitution, a move meant to show the change is permanent and binding for domestic and global use.More news: Federal judge temporarily blocks deportation of pizza worker
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Significantly, they agreed that the new name would be used both internationally and bilaterally, so that even the 140 or more countries that recognize the name Macedonia will also have to adopt North Macedonia. "Our bid in the compromise is a defined and precise name, the name that is honorable and geographically precise - Republic of Northern Macedonia". It also has to be ratified by both Macedonian and Greek parliaments.
"In other words, if the constitutional amendment is not successfully completed, then the invitation to join North Atlantic Treaty Organisation will be automatically rescinded and the accession talks with the European Union will not start", he said.
NATO Secretary General Jens Stoltenberg said Tuesday's "historic agreement" was "testament to many years of patient diplomacy", and called on the two countries' prime ministers to finalize the deal.
However, Mr Tsipras said, this will be contingent on Macedonia completing the constitutional changes. I believe in the democracy of the internet and want to keep this site and its enriching content free for everyone.
After the final agreement has been reached, a joint statement for the Greek and Macedonian public is expected to be published.
In Athens, Greek Prime Minister Alexis Tsipras called the deal "historic".
Greek opponents of the deal say it would not go far enough.
The dispute has poisoned relations between the two neighbors since Macedonia's independence from Yugoslavia in 1991. "We just saw a press conference where the defeat is shown as a fake victory", Hristijan Mickoski, president of opposition party VMRO-DPMNE said.