Germany has strenuously opposed the issue of joint debt — or eurobonds — because, while it would immediately ease pressure on countries like Spain, German taxpayers would be put on the hook for foreign debts and Germany's cost of borrowing would increase.
Asked in Luxembourg what Germany would think of her suggestions, Lagarde smiled and said "We hope wisdom will prevail."
In Rome, Hollande said eurobonds need to remain a possibility "and not in 10 years," without specifying a timeframe. Rajoy spoke favorably about Monti's proposal to use bailout funds to buy bonds of vulnerable countries like Italy and Spain on secondary markets.
The leaders said they would spend the coming days lobbying their EU counterparts.
A growing number of international leaders have called on Merkel and the eurozone to find quickly a comprehensive solution to the debt crisis rather than continuing to take piecemeal measures that provide only temporary relief. At this week's G-20 summit of world economic powers in Los Cabos, Mexico, politicians, including U.S. President Barack Obama, called on Europe to do what was necessary.
Speaking at an economic forum in St. Petersburg, David Lipton, first deputy managing director at the International Monetary Fund, urged European leaders to act quickly: "The markets are beginning to question the viability of the European monetary union itself. It's very important that eurozone countries address the long-run question: Where is the architecture of the European monetary union going?"
The meeting was moved up by a few hours to give Merkel time to fly to Poland to watch Germany play Greece in a European Championship quarterfinal match. The game pits the teams of Europe's strongest economy, Germany, against the eurozone's most troubled, Greece. The just-installed Greek leader had no immediate plans to attend.
Barry reported from Milan. Geir Moulson contributed from Berlin and Nataliya Vasilyeva contributed from St Petersburg.
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