The election, which coincides with a key local vote, is seen as a test for Prime Minister Antonis Samaras' fragile coalition in a country still reeling from the debt crisis and banking on loans from the European Union and International Monetary Fund.
Samaras' New Democracy - which leads the coalition with the Socialist PASOK party - and its rival, the leftist Syriza party, have been running neck-and-neck in polls for months, with voters often swinging in favor of one or the other party.
But New Democracy has seen a boost in ratings since Athens tapped bond markets in April for the first time in four years and Samaras promised to spend a 527 million euro ($731 million) windfall from Greece's 2013 budget surplus on poor, austerity-hit Greeks.
The poll conducted on April 28-30 by Metron Analysis for Sunday's Ethnos newspaper put support for New Democracy at 28.4 percent and for Syriza at 27.6 percent, excluding undecided voters - who otherwise stood at 17 percent.
In a poll by the same agency in April, Syriza was leading with 1.4 percentage points.
To Potami, a movement launched in February by a popular TV journalist, ranked third with 10.9 percent. It was followed by the Communist KKE party with 6.6 percent.
The far-right Golden Dawn, which entered parliament for the first time in 2012 tapping into popular anger with austerity and rising unemployment, was in fifth place scoring 6 percent.
Syriza has called the vote a referendum on austerity and the EU/IMF bailout and wants to push for early parliamentary elections.
About 60 percent of the respondents in the Metron Analysis poll said they did not want early elections.
An MRB poll for Star TV, which was broadcast on Friday, also showed New Democracy leading with 0.8 percentage points. Syriza was ahead with 0.2 percentage points in a previous MRB poll.