FRANKFURT—The European Central Bank’s governing council was meeting on Sunday to decide whether to keep disbursing emergency loans to Greek banks, a decision that could set off a period of turmoil in the country as it prepares for a bailout referendum next weekend.

The ECB faces a choice between maintaining ample liquidity support for Greek banks and potentially undermining the ECB’s credibility as a rules-based institution, or forcing the closure of Greek banks and enraging Greece’s government and people.

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