BRUSSELS—Eurozone officials shook their heads when Yanis Varoufakis, Greece’s mercurial new finance minister, repeatedly told journalists after the Greek elections last month that his government was “insolvent” and “bankrupt.”

At meetings of eurozone finance ministers on Monday night and last week, European Central Bank President Mario Draghi warned Mr. Varoufakis that such statements threatened to accelerate the flight of deposits from Greek banks, officials present at the meetings said.

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