The main Irish political parties back the treaty but the "No" campaign has mounted a challenge, the BBC reported.
Parliaments of the other EU members decided the matter, but Ireland, by law, must conduct a popular vote on changes to its constitution.
The treaty, known as the Lisbon Treaty, would remove the national veto in more policy areas, provide for a new president of the European Council and strengthen the foreign affairs post.
Dick Roche, Ireland's EU minister, said the results would be "very, very close."
The "No" campaign is a coalition of organizations, including Sinn Fein, the only party in Parliament opposing the pact.
Sinn Fein leader Gerry Adams said a successful "no" vote would give Ireland's government "a strong mandate to negotiate a better deal for Ireland," the British broadcaster reported.
European Commission President Jose Manuel Barroso urged EU members to back the treaty, which would go into affect in January.
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