Hungary's centre-right opposition party looks set to become the first party to single-handedly secure a two-thirds majority in a general election.
Viktor Orban's Fidesz party won 52.7 per cent of the votes and 206 parliamentary seats in the first round on April 11.
It has a strong chance of obtaining the 52 seats it needs for the two-thirds majority from the 121 seats to be decided on Sunday.
In the event, Orban is expected to become Hungary's next prime minister.
A win would enable Fidesz, last in power between 1998 and 2002, to implement reforms such as streamlining the bloated local government system and making dual citizenship easier to get for millions of ethnic Hungarians in neighbouring countries.
Fidesz, running on a joint ticket with the small Christian Democrats, now stands a chance of forming the first non-coalition government with a two-thirds mandate in Hungary's 20-year post-communist history.
"We now have the opportunity to create an unparalleled unity and, with it, initiate changes unprecedented in both their scale and swiftness," Orban told Hungary's Magyar Nemzet newspaper on Friday.
"The entire country feels that it can now make history and this sentiment is a tremendous driving force."
The last time any government secured a two-thirds mandate was in 1994 when the Socialists teamed up with the liberal Free Democrats.
The other parties with seats in the next Hungarian parliament already are the Socialists with 28, far-right Jobbik with 26 seats, and Green LMP, which has five seats.
More jobs pledged
Fidesz has pledged more jobs, less bureaucracy and lower taxes to revive the economy. But with a history of high deficits and the budget under the microscope of international lenders, the room for fiscal stimulus will be limited.
At a news conference after the first round of voting, Orban promised to "put Hungary on the path of economic growth", pledging "substantial tax cuts already by this year".
Hungary was one of the countries hardest hit by the global recession and had to be bailed out to the tune of 20bn euros by the International Monetary Fund, the World Bank and the European Union. That rescue package came with stringent conditions attached.
Even Laszlo Kover, Orban's closest collaborator, said "it is clear that taxes won't be cut this year, but towards the end of our four-year term".
Polls will close at 1700 GMT and results expected around 1900 GMT.