The disagreement hinges on who should be involved in deciding how to tackle the eurozone's problems.
European leaders are due to hold a crucial summit on Wednesday, at which Germany and France have said they want to reach a breakthrough.
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Initially, only representatives from the 17 countries that use the euro were invited to attend - but the UK is insisting that all 27 EU members have a stake in the discussions.
At preliminary talks in Brussels on Sunday, Prime Minister Cameron said:
"We must safeguard the interests of countries that want to stay outside the euro, particularly with respect to the integrity of the single market for all 27 countries of the EU."
President Sarkozy is quoted as responding angrily to the UK's position:
"You have lost a good opportunity to shut up.
"We are sick of you criticising us and telling us what to do. You say you hate the euro and now you want to interfere in our meetings."
Eventually leaders agreed that all 27 members will debate the crisis plan, but the 17 eurozone countries will have the final say, reports the Guardian. Cameron nonetheless secured an agreement that a level playing field would be safeguarded for countries not in the euro.
A second meeting is planned for Thursday so that non-euro countries can "rubberstamp" the decisions made on Wednesday, according to Der Spiegel.
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Sarkozy and Cameron's spat came the day before the UK parliament is due to debate whether to hold a referendum on Britain's membership of Europe.
Cameron has instructed members of his Conservative party not to vote for a referendum, but several MPs are threatening to rebel. They claim the majority of British Conservative voters want to renegotiate the UK's membership of Europe to give it greater say over its national interests.