PARIS — Cyprus on Monday said it was formally requesting a bailout from the European Union in a bid to bolster its struggling banks, making it the fifth euro zone country to request a rescue.

The announcement came after weeks of concern that the crisis in Greece and a potential Greek exit from the euro could bring down the economy in the small island nation, whose banks are heavily exposed to Greece.

The slim victory of a pro-euro party in recent Greek elections helped to temporarily alleviate that concern, even as economists said Cyprus could need as much as €10 billion, or $12.5 billion, to shore up its ailing banks and cash-strapped public sector.

Government officials would not specify how much Cyprus would be seeking from the E.U.’s bailout fund, saying negotiations were continuing. Leaders of the Union were scheduled to meet Thursday and Friday in Brussels, where they are expected to discuss the terms of Greece’s bailout.