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By Andras Gergely

BUDAPEST, June 9 (Reuters) - Hungary's prime minister said on Saturday he would press on with reforms in health care and education, and that his Socialist Party were closer to resolving differences with their coalition partners on the reforms.

Ferenc Gyurcsany has introduced tax and price hikes to cut Hungary's huge budget deficit, but his party has not been able to agree with the smaller Alliance of Free Democrats (SZDSZ) on how to reform the inefficient and costly health care system. "We will not give up and I will also not give up," Gyurcsany told a party congress to mark the first year since his government's re-election.

The Free Democrats are pressing Gyurcsany to speed up the reforms as the price for their staying in the government. "We are partners in reform. However, the results have been mixed," Free Democrat president Janos Koka told daily, Nepszabadsag.

The liberal Free Democrats want to introduce multiple insurers in health care provision. But some MPs on the left of Gyurcsany's Socialists say competition will lead to poor people not getting proper cover, and fear their leader is giving too much ground to free marketers.

"Lots and lots of consultation is needed. But at the end there will be an agreement, we are closer to that today than ever," Gyurcsany told the daily Nepszava in an interview.

The SZDSZ wants to agree with the Socialists by the end of June on a timetable for further reform, which must include health, education and taxes, news agency MTI quoted Koka.

The liberals also want to hold talks on church financing and reforming the police force and welfare institutions, he said.

Asked whether the SZDSZ would be prepared to support the government from outside, Koka, who is economy minister, said:

"We are thinking in terms of the conditions of support from inside."

Gyurcsany has won praise from the EU for his reform measures, but has alienated voters by admitting he lied to win re-election in 2006.

Some recent opinion polls have put the Socialists on less than 20 percent and the Free Democrats on less than 5 percent.

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