The various parts make up the EU's so-called banking union, an effort to make sure that a bank failure in one country does not overwhelm an individual state's finances, threatening the wider region's stability.
However, the ministers were still far from reaching an agreement on how to design and fund the bank rescue authority.
Germany and other countries that have paid the bulk of Europe's bailouts have concerns about the institution's legal basis and fear their taxpayers will be stuck with bills to clean up banks in Europe's weaker economies.
Luxembourg Finance Minister Luc Frieden argued the 17-nation eurozone needs a credible financial backstop since most banks operate across borders.
Michel Barnier, the EU Commissioner in charge of financial sector reforms, added that the new banking supervisor cannot work if there isn't alongside it a bank rescue authority that can take the actions needed to keep the financial system stable.
"A banking union also requires action to restructure non-viable banks when necessary," he said.
European officials are determined to spare taxpayers from having to pay for further bank bailouts, and plan to rely instead on a levy to be paid by banks to build up the backstop fund. That, however, will take years, maybe even decades. Some officials therefore argue the bank backstop fund should be able to borrow money in the meantime.
But German Finance Minister Wolfgang Schaeuble, representing Europe's biggest economy, quickly rejected that idea.
The EU hopes to reach an agreement over the next two or three months to ensure the legislation can make its way through the European Parliament, whose term ends in May.
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