But the agreement has faced vocal opposition on both sides of the border.
Opponents in Macedonia include the country's president, Gjorge Ivanov, who has called the deal a "flagrant violation of sovereignty." They have called for a boycott of Sunday's referendum.
Voters were confronted with the question: "Are you in favor of membership in NATO and European Union by accepting the deal between (the) Republic of Macedonia and Republic of Greece?"
The referendum was called as a consultative, non-binding move. The distinction means the government could take the outcome as a fair reflection of public opinion and act accordingly, regardless of the turnout. Under the country's constitution, a binding referendum would need a minimum turnout of 50 percent to be considered valid.
The campaign has been relatively muted, with most posters favoring the government-led "Yes" campaign.
Supporters of the deal, led by Prime Minister Zoran Zaev, have focused on the vote as being the lynchpin of the country's future prosperity, the key to its ability to join NATO and, eventually, the European Union. It would be a major step for a country that less than two decades ago almost descended into civil war, when parts of its ethnic Albanian minority took up arms against the government, seeking greater rights.
Djose Tanevski was among the early voters Sunday in Skopje, the capital.
"I came here because of the future of our children, who should have a decent life, a life in a lovely country, which will become a member of the European Union and NATO," he said.
If the "Yes" vote wins, the next step is for the government to amend parts of the country's constitution to ensure it doesn't contain anything that could be considered irredentist against Greece. Only after those changes are approved by parliament does the deal face ratification in Greece.
The referendum has stirred strong interest in the West. German Chancellor Angela Merkel and U.S. Defense Secretary Jim Mattis have been among the top foreign officials heading to Skopje recently to urge its people to vote "Yes."
There's been growing concern over the reach of Russia, which is not keen on NATO expanding in a part of Europe that was once within its sphere of influence. Mattis said there was "no doubt" that Moscow funded groups inside Macedonia to campaign against the name change.
Even if Macedonians vote in favor of the deal, the agreement still faces several hurdles before it can be fully ratified.
The constitutional amendments that are required need a two-thirds majority of parliament's 120 members to go through. So far Zaev has pledges of support from 73 — seven short of the required number. A low turnout on Sunday could complicate his task in persuading more lawmakers that the name change agreement with Greece reflects the will of the people.
Once the hurdle of constitutional amendments is overcome, Greece must then ratify the deal.
But Greek Prime Minister Alexis Tsipras faces problems of his own. His governing coalition partner, right-wing Independent Greeks head Panos Kammenos, has vowed to vote against the deal in parliament, leaving Tsipras reliant on opposition parties and independent lawmakers to push the deal through.