Around 200 of the biggest banks will come under the direct oversight of the European Central Bank, which will act as chief supervisor of eurozone banks.
The agreement – a key step towards banking union – will be put before European leaders later on Thursday.
New rules on prudent banking are seen as vital to bolster the euro, as bank failures triggered the financial crash.
The measures are also aimed at preventing banking failures ending up on the books of eurozone governments.
“We have reached the main points to establish a European banking supervisor that should take on its work in 2014,” said German Finance Minister Wolfgang Schaeuble, after 14 hours of talks ended shortly before dawn on Thursday.
“Piece by piece, brick by brick, the banking union will be built on this first fundamental step today,” said EU Commissioner Michel Barnier.
German Chancellor Angela Merkel welcomed the agreement, telling the Bundestag (lower house of parliament) that Germany’s “core demands” had been secured.
“It cannot be praised too highly.”
She has previously warned against rushing into banking union out of concern that Germany would face further financial demands.
Significantly, a large number of French banks will be supervised by the ECB but rather few institutions in Germany will, because of its fragmented banking industry.
European Commission President Jose Manuel Barroso hailed the deal as “a crucial and very substantive step towards completion of the banking union”.
UK Chancellor George Osborne said the aim of protecting the interests of EU states not signing up to the banking union “has been achieved”.
Under the deal, banks with more than 30 billion euros ($39 billion) in assets will be placed under the oversight of the European Central Bank.
The ECB would also be able to intervene with smaller lenders and borrowers at the first sign of trouble.
Europe’s finance ministers have taken another major step towards closer integration, with a significant transfer of authority from national governments to the ECB.
The EU had already agreed that the ECB would act as chief supervisor of eurozone banks.
But the deal gives the ECB powers to close down eurozone banks that do not follow rules. It also paves the way for the EU’s main rescue fund to come to the direct aid of struggling banks.
It represents the first stage of a banking union – known as a Single Supervisory Mechanism (SSM) – which EU leaders believe can be put in place without having to change EU treaties.
But there have been some legal doubts about the subsequent stages – a joint deposit guarantee scheme and a joint resolution mechanism for winding up broken banks.
The UK, which is not in the eurozone, will not be joining the banking union but has won some protection against being marginalized when key decisions are taken.
The UK and Denmark both have formal opt-outs from the euro.
The other EU states still outside the euro are committed to joining, and can sign up to the banking union in the meantime, although Sweden and the Czech Republic have made clear they will not.
For months, the scope of the ECB’s supervisory powers was the subject of strained negotiations.
France and Germany had been unable to agree the threshold at which the ECB would intervene – with Germany arguing that many of its regional banks were too small to warrant ECB attention.
While the European Central Bank will be responsible for the overall running of the SSM, it will be in close co-operation with the supervisory authorities of member states and the EU-wide European Banking Authority, which creates banking rules across all 27 member states.
The summit’s chairman, European Council President Herman Van Rompuy, will try to get a commitment to launch the SSM in January 2014 at the latest. His vision for far-reaching eurozone integration is set out in a report, which will be the focus of the discussions among EU leaders in Brussels on Thursday.
While banking union is the immediate focus, the report also proposes “contractual” arrangements between eurozone governments and the Commission, to prevent governments delaying, or reneging on, important economic reforms.
EU leaders are likely to avoid any measures that could trigger treaty change before the European elections in mid-2014, because treaty change is nearly always a thorny issue for the EU. It took seven years for the EU to adopt the Lisbon Treaty.
There is strong opposition in Germany and other richer eurozone nations to any further taxpayer-funded bailouts of indebted banks and governments.
Germany’s Constitutional Court has already flexed its legal muscles over eurozone integration.
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