The election raises fears that a Greek exit from the euro could spread contagion to other eurozone members and put the global economy into turmoil.
The election, the second in six weeks, was called after a vote on May 6 proved inconclusive. with no party or coalition able to form a government.
The head of New Democracy, Antonis Samaras, broadly accepts Greece's international bailout, but says he will renegotiate the terms of the agreement to seek a better deal.
Many Greeks are unhappy with the austerity conditions.
Samaras also said he could lead a coalition of other centre-right parties and the Socialist Pasok party.
Samaras' main rival is Alexis Tsipras, the youthful head of the left-wing Syriza party.
Meanwhile, The Guardian reported European leaders were seeking to calm fears that the Greek election would trigger a run on eurozone banks after polls showed Syriza could deny pro-austerity parties overall control in the election.
British Prime Minister David Cameron joined German chancellor, Angela Merkel, and French president, Francois Hollande, in emergency talks to prepare for the election aftermath. A
-Agencies