Slovakia's support is critical to boosting the bailout fund, as it is one of several countries in which parliamentary approval is needed to ratify an agreed increase in the European Financial Stability Facility's lending capacity.
It will vote after all other euro zone member states, and has not given a firm date for the vote. A "no" vote could threaten the entire EFSF deal.
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Euro zone leaders agreed on July 21 to allow the 440-billion-euro EFSF to give precautionary loans to countries under attack in the markets and to buy sovereign bonds, in exceptional circumstances, to prop up struggling states. But many countries face hurdles in convincing skeptical parliamentarians at home to back the pledge.
In Slovakia, the single currency area's second poorest member, the junior government Freedom and Solidarity party (SaS) remains adamantly against EFSF strengthening and the Greek rescue program.
But Sulik, speaker of parliament, said the ruling coalition of Prime Minister Iveta Radicova will not split because of his party's reluctant stance.
"The euro and the euro zone are great projects, if only the rules would be followed," Sulik said. "The euro's biggest threat is the EFSF."
He slammed the European Central Bank's (ECB) decision to buy Italy's sovereign bonds on the secondary market, aimed at easing pressure from financial markets after one of the euro area's major economies landed at heart of the spiraling debt crisis.
"Italy must start saving, it is outrageous that the ECB is buying Italian bonds, this is breaching all possible rules," Sulik said, adding Italy should start selling its gold reserves to cut its debt burden.
Radicova, unless she succeeds in persuading Sulik to back the measures, will need to seek the support of the strongest opposition party SMER in order to ratify the new EFSF powers.
But the center-left SMER, led by ex-Prime Minister Robert Fico, said if Radicova's coalition secured joint support for the EFSF bills, his pro-euro party would vote in favor. If not, they would have to enter into negotiations.
"Either you secure enough votes, or you will have to sit down with us and talk," Peter Kazimir, SMER's vice-chairman and former deputy finance minister said in a TV debate on Sunday.
(Reporting by Martin Santa; Editing by Rosalind Russell)