Italy, Belgium and others argued more flexibility is needed so they can boost the economy and support job creation, but the more hawkish northern European nations including Germany rejected the idea.
Belgian Prime Minister Elio di Rupo said at a meeting of EU leaders there needs to be "more flexibility, specifically for certain countries seeking to balance their budgets a few years later."
"Growth needs to be restarted to avoid hurting companies and citizens," he added.
But Finland's new Prime Minister Alexander Stubb said stability was paramount to avoid another financial crisis. He insisted the rules "need no more flexibility."
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The debate comes as EU countries are under less pressure from financial markets because they have seen their government borrowing rates fall. Investors have regained confidence in the 18-country eurozone as its financial crisis eased. But while markets have stabilised, economic growth is still weak and unemployment is stuck at record-levels in many of the bloc's nations.
Italian news reports said Renzi and German Chancellor Angela Merkel, who opposes changing the rules, had a tense exchange at the leaders' dinner yesterday.
A senior German government official, speaking on condition of anonymity, said the two leaders had a more positive bilateral meeting this morning. The official was briefing reporters on condition of anonymity because he was not allowed to discuss their closed-door talks publicly.
The EU rules limit countries' budget deficit to a maximum of 3 per cent of gross domestic product and the total debt to 60 per cent, forcing countries above the thresholds to consolidate their budgets.