ATHENS (Reuters) - Greece's ruling conservatives have grabbed a narrow lead over the leftist main opposition since securing bailout funds to avert bankruptcy, three opinion polls showed, as most Greeks believe the country will stay in the euro zone.

A survey conducted by MRB pollsters for Sunday newspaper Real News showed that if elections were held now Prime Minister Antonis Samaras's New Democracy party would get 29.2 percent versus 27.8 percent for the anti-bailout SYRIZA party.

The ultra-nationalist Golden Dawn ranked third with 11.6 percent while the power-sharing Socialist PASOK party stood at 7.0 percent, the poll conducted in January 22-24 showed.

For months New Democracy trailed Syriza in polls but it pulled ahead this month after securing aid from the European Union and the International Monetary Fund to end months of uncertainty over Greece's future in the euro.

"There is an increase in the number of people who believe that we must stay in the euro zone and stick to our commitments because there is no other way out," said MRB's Dimitris Mavros.

MRB said that for the first time since the Greek crisis broke out a majority of Greeks - 60 percent - said they believed the country would not go bankrupt and more than 75 percent said they expected Greece to stay in the euro.

But as many as 83 percent expected more austerity measures.

Another survey conducted by Alco for newspaper Proto Thema in the same period showed New Democracy moving into first place for the first time since national polls in June. If elections were held now it would get 24.9 percent versus SYRIZA's 23.4 percent, Alco showed.

Questions still remain over whether the coalition can continue to implement a reform program that has sent unemployment to the highest level in the European Union.

Greeks fed up with waves of tax hikes and salary cuts have repeatedly taken to the streets in often violent protests and lashed out at politicians they hold responsible for the crisis.

More than 62 percent polled by Alco said they could not afford to pay off their loans and their taxes.

But a Kapa Research survey for Sunday's To Vima newspaper showed that 57.6 percent believe the government deserves more time against 28.9 percent who want early elections.

(Reporting by Renee Maltezou)