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Decisions on banking supervision, how to use euro zone bailout money, aid to Spain and Cyprus and whether to grant concessions to Greece are likely to take months to finalize, while pressure for action is growing.

Spanish and Italian borrowing costs moved back up near unsustainable levels on Friday as hopes raised by the summit began to fade. Leaders of both countries issued pleas at the weekend for rapid moves to implement the agreement.

"This is where the credibility of the entire European project is at play," Spanish Prime Minister Mariano Rajoy said.

The deal reached by leaders from the 17 nations sharing the euro aims to give the European Central Bank greater oversight of the bloc's banks and to use the euro zone's rescue funds to reduce countries' borrowing costs.

But critical elements were left vague, time frames may already be slipping and ministers could end up meeting again later in July to take firm decisions.

"This is very much the follow on from the summit, but it doesn't mean all details can be set down," said one euro zone diplomat briefed on the Brussels meeting that is due to start at 11. a.m. EDT.

"The issue of ECB supervision is a complex, longer-term issue and not one that can be decided in a few hours."

ECB President Mario Draghi will testify to the European Parliament on Monday before ministers meet, and after cutting rates last week could signal more dramatic measures such as buying government bonds or flooding banks with fresh liquidity.

Germany, the bloc's biggest economy, as well as wealthy Finland and the Netherlands, are wary of what was announced at the summit and German Chancellor Angela Merkel is reluctant help its partners without strict conditions.

Central to the euro zone leaders' plan is to give the ECB a central role in supervising banks, which would then allow the permanent rescue fund - the European Stability Mechanism (ESM)- to recapitalize banks directly instead of via governments.

That is seen as a concession to Spain, which has requested a bailout of up to 100 billion euros ($125 billion) for its banks, although it is not clear when Madrid will benefit.

Leaders want to break the link between banks and sovereigns by not lumbering governments with debts for rescuing their lenders, making it harder for them to borrow.

But the central question as to whether individual countries or the euro zone assumes liability for banks that are rescued by the ESM remains open.

Leaders agreed to remove the ESM's preferred creditor status when it lends to Spain, to calm investors who were worried they would not be repaid money they had already lent.

They also decided that the ESM and the euro zone's temporary bailout fund, the EFSF, can buy euro zone bonds at auction and in the open market to lower borrowing costs, with some conditions attached but without a full program.

The ESM is due to start operating during the European summer, but at least for now, countries will need to provide guarantees in return for bank aid it gives, according to one euro zone official who is involved in preparing the Eurogroup.

That might help overcome German concerns about the ESM taking on this risk.

"There is some degree of mystification going on here ... in the broader public who think that under current rules the ESM could all of a sudden end up owning Bankia with the full risk of Bankia on the balance sheet of the ESM," he said, referring to the Spanish lender. "This is very much not the case."

OPTIMISTIC

As always in the euro zone's crisis management, finance ministers are given the difficult task of juggling national interests, in particular among the bloc's four biggest economies, Germany, France, Italy and Spain.

But it looks optimistic that they can do what leaders said in their statement on June 29 when they told the Eurogroup of finance ministers "to implement these decisions by July 9".

Much depends on the ECB's crucial role as supervisor, which will need to be grounded in European law. It falls to the European Commission to propose such legislation, which is not expected until at least September.

Despite the obstacles to the broad package outlined by leaders, the range of measures agreed allow some short-term action, and vocal opposition to euro zone bond buying in the Netherlands and Finland is unlikely to ruin those plans.

Finland has said it opposes bond-buying in secondary markets, because it considers such purchases to be ineffective.

In emergency cases, the ESM's treaty allows for decisions to be taken with an 85 percent majority, and the Netherlands and Finland only account for 8 percent combined.

Coordinating euro zone finance ministers has been the job of Luxembourg Prime Minister Jean-Claude Juncker since 2005, but his terms ends on July 17 and ministers are due to discuss his successor on Monday. However, confusion over who that is likely to be means his term may be extended.

GREECE MUST DELIVER

Ministers will discuss the findings of the "troika" of the European Union, the European Central Bank and International Monetary Fund from their first mission to Greece since the June 17 election. Another mission is due to return later in July.

Greece's new Finance Minister Yannis Stournaras said on Thursday he had been warned to expect a tough time at the Eurogroup, having acknowledged Athens was off course on its pledges linked to a 130-billion-euro rescue.

One senior euro zone official said the Eurogroup needed to see that Athens is getting back on track before it can hand over more aid, even if the previous Greek government said the administration risked running out of money by the end of July.

Greece's Prime Minister Antonis Samaras wants to ease the terms of the bailout, but that would mean more money for Athens.

"Even if the second program as it stands were fully implemented, it is not clear that market access could resume (in 2015)," said David Mackie, an economist at JP Morgan. "A third program seems likely in any event."

For Spain, ministers are likely to agree in principle to an aid package although no formal green light will follow until the end of the month, one euro zone official said.

(Writing by Robin Emmott; editing by Philippa Fletcher)