European Union finance ministers outlined a deal on Saturday for recapitalizing European banks, and the leaders of Germany and France said they hoped for a breakthrough in tackling the Eurozone debt crisis at a summit on Wednesday.

After nearly 10 hours of talks, finance ministers overcame strong opposition from Italy, Portugal, and Spain as they agreed on the need to inject around 100 billion euros into European banks to protect them from the threat of a Greek debt default and the broader risks of financial contagion in the Eurozone.

The ministers will submit their thoughts to EU leaders, who meet on Sunday to discuss a comprehensive solution to the debt crisis, which needs to contain a second bailout program for Greece, a scaling up of the Eurozone's bailout fund, and the strengthening of European bank balance sheets.

No headline deal is expected from Sunday's meeting, but German Chancellor Angela Merkel said she was hopeful that another Eurozone summit scheduled for Wednesday would produce definitive results, and French President Nicolas Sarkozy agreed.

We have to take far-reaching decisions, Merkel told reporters ahead of a presummit meeting near Brussels. I believe that the finance ministers made progress, so that we can achieve our ambitious targets by Wednesday.

Speaking to journalists in Brussels, Sarkozy said: Progress has been made. Between now and Wednesday, a solution must be found, a structural solution, an ambitious solution, a definitive solution. Asked if he was confident that could happen, he replied: Yes, otherwise, I wouldn't be here.

During their meeting on Saturday, EU finance ministers heard from the head of the European Banking Authority, who told them that if EU banks were to raise their core capital ratios to 9 percent, and if the bad government bonds on their books were accounted for at current prices, then between 100 billion euros and 110 billion euros ($138.9 billion and $152.8 billion, respectively) was needed to shore up the banking system.

Italy, Portugal, and Spain would face paying hefty prices to strengthen their banks, so they were reluctant to agree to a deal that they see as putting them more in the firing line than France and Germany, which also have large exposures to Greek debt.

But under intense pressure from the other 24 EU states, the outlines of a deal were agreed on, officials said. However, sources said that the proposal EU leaders receive from finance ministers on Sunday may not mention a recapitalization figure, leaving that for the leaders to haggle over.

We have laid down the foundations for an agreement, said Swedish Finance Minister Anders Borg as he left the meeting, a position seconded by British Chancellor of the Exchequer George Osborne.

If EU leaders are able to reach a deal on bank recapitalization in the coming days, it would be a significant step forward in efforts to contain a crisis that has raged for nearly two years and threatens the EU and global economy.

But several major areas of disagreement remain, and it will require vast amounts of hard negotiation between Sunday and Wednesday to strike a deal that convinces financial markets and Europe's major trading partners that the crisis is in hand.

The biggest sticking point is agreeing on how best to scale up the European Financial Stability Facility, the 440 billion euro emergency fund set up last year and so far used to bail out Ireland and Portugal.

Financial markets are not convinced the EFSF is big enough to handle the threat of deeper bond-market turmoil in Italy and Spain, so leaders are examining ways they can raise the EFSF's firepower without increasing their commitments to the fund.

One proposal is to the use it to provide guarantees to buyers of Italian, Spanish,and other at-risk Eurozone debt in an effort to convince institutions the bonds are safe to buy.

By guaranteeing only a portion of the bonds -- say, 20 percent -- the EFSF could reach up to five times further.

However, France and several other member states believe a better approach would be to turn the EFSF into a bank so that it could access European Central Bank funds, potentially providing it with unlimited liquidity.

Germany, the Netherlands, Finland, and the ECB are opposed to the idea, which the European Commission also says would probably violate EU treaty rules.

Another significant problem is what to do about Greece, the country whose spiralling debts and efforts to cover up its economic shortcomings first provoked the crisis.

Greece has already received a 110 billion euro rescue plan from the EU and International Monetary Fund, but now needs another bailout.

On July 21, a deal was struck in which Greece's private-sector bondholders would voluntarily accept a 21 percent reduction in the value of their bonds in an effort to lighten Athens' debt burden. At the same time, the EFSF and IMF agreed to provide a further 109 billion euros of aid.

But that deal has unravelled as Greece's economic situation has deteriorated, with budget deficit targets missed and growth contracting more deeply. Now the private sector may need to take a 50 percent writedown on its bond holdings, and the public sector may need to provide substantially more.

Back to Greece

While Greece is one of the Eurozone's smaller economies, finding an acceptable way of rescuing it is proving ever more intractable. Represented by the Institute of International Finance, the private sector is deeply opposed to a renegotiation of the July 21 deal, concerned that a deeper writedown will force some banks into very severe losses.

On Saturday, its managing director, Charles Dallara, who has been attending talks in Brussels, said that progress was being made, although it was limited.

We remain open to explore options on a voluntary approach built on a realistic outlook for the Greek economy and restoration of Greece's market access, he said.

Since French banks are among the largest holders of Greek debt, Paris is reluctant to push the private sector too far, and wants to ensure any deal remains voluntary. At heart, there is concern about France losing its triple-A credit rating if some of its banks end up needing recapitalization by the state.

At the same time, Eurozone leaders are applying increasing pressure on Italy and Spain to do much more to get their budget deficits in check and stimulate leaden growth.

With yields on Italian and Spanish bonds being pushed to near-record level highs on financial markets, the cost of funding their deficits has risen dramatically, creating liquidity problems that the Eurozone can ill afford.

EU leaders want to press Italian Prime Minister Silvio Berlusconi in particular to stick to his commitments on spending cuts and investment measures to spur growth, in the hope of better insulating Italy against market pressure.

In a speech to members of her Christian Democrat party on Saturday, German Chancellor Merkel urged countries like Italy and Spain to reduce their sovereign debt levels.

Spain has already done a lot, but it will probably have to do more to win back the markets' confidence, Merkel said.

If they don't do anything with their budgets, if they continue to have [debt] equal to 120 percent [of gross domestic product] like Italy, then it won't matter how high the protective wall is because it won't help to win back the markets' confidence.

(Additional reporting by Matthew Falloon, Andreas Rinke, Ilona Wissenbach, Daniel Flynn, Annika Breidthardt and Robin Emmott; Writing by Sebastian Moffett and Luke Baker; Editing by Jon Boyle)