Mr Tsipras earlier Sunday accepted the resignation of Defence Minister Panos Kammenos, who is head of his main coalition ally in government, the nationalist Independent Greeks party (ANEL).
The baffling week that followed the New Year and Orthodox Christmas holiday was marked by conflicting statements and signals from key Greek political figures as developments in Greece moved into high gear immediately after the Macedonia/FYROM Parliament approved the constitutional amendments required by the Prespes Agreement.
The coalition government of SYRIZA and the Independent Greeks (ANEL) party, in power since January 2015, had been widely expected to fracture over the parliamentary ratification process for the Prespes Agreement.
Now Tsipras needs only a majority in the 300-seat Greek parliament to ratify the accord. However, the Greek Parliament still needs to vote - and this will likely prove no easy task given how deeply divisive the issue remains there, a report in The New York Times observed.
ANEL leader Kammenos has long promised to reject the Prespes Agreement but several of his MPs have said publicly they will support it, generating new internal break-up scenarios each day.
The two countries recently agreed that Greece's northern neighbour would be called North Macedonia, therefore ending a 27-year-long row.
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ANEL is the junior party in Greece's ruling coalition and strongly opposes the name change, a rift that threatens to destabilise the government of Prime Minister Alexis Tsipras, who helped broker the deal.
The deal struck by Mr Tsipras and Mr Zaev came after a change of the Government in Macedonia in 2017, which led to a more distended relationship.
Panos Kammenos announced his resignation after meeting with Prime Minister Alexis Tsipras on Sunday morning.
Tsipras has said a separate vote on a protocol to enable Macedonia to join North Atlantic Treaty Organisation would be held "not long afterwards".
Kommersant estimates that the adoption of constitutional changes can have another effect, important not only for that Balkan state, adding that "in the opinion of regional experts, this can influence the granting of autocephaly to the Macedonian Orthodox Church". "But I am convinced that the Greek parliament will also find the strength to make the decision", he said.
Present-day Macedonia and northern Greece were part of a Roman province called Macedonia and both claim the heritage of Alexander the Great two millennia earlier.
New Democracy is vehemently against the agreement but Tsipras hopes to convince opposition MPs from smaller parties to back the historic name change deal. Greek parliamentary endorsement of the name is required for the tiny Balkan nation to join the European Union and North Atlantic Treaty Organisation.