Embattled Greek banks were hit with more damaging data Friday that showed savers withdrew funds at a rapid clip.

Deposits at Greek banks are at their lowest level in more than 10 years amid broad concerns about the country's economic prospects that have hammered shares in Greek lenders this year.

Total deposits fell to €139.36 billion ($153 billion) in April, down from €145.04 billion in March and over €170 billion just five months ago, data from the European Central Bank showed.

Shares in the country's biggest banks—National Bank of Greece SA, Piraeus Bank SA, Eurobank Ergasias SA and Alpha Bank AE—were mixed on Friday. So far in 2015, an index of Greek bank stocks is down more than 27%, according to FactSet. By comparison, the Stoxx Europe 600 subindex of European banking stocks has risen more than 17% and Greece's benchmark stock index is up around half a percent.

David Vickers, a senior portfolio manager at Russell Investments, said that it would take a more significant outflow of deposits to drive a steep downturn in bank shares.

"It looks like another 30% fall is deposits is needed before the banks run out of collateral—I'm presuming the market expects a deal to be done prior to that situation evolving," he said.

The country's banks could be battered further if Greece doesn't reach a financing deal with its international creditors in coming weeks, economists say. But some high-profile investors have backed Greek banks in recent months.

Hedge-fund manager John Paulson is still sanguine on Greece. In mid-April, the American billionaire's Paulson & Co. Inc. held 132 million shares in Piraeus, according to Thomson Reuters, making him one of the bank's top three shareholders.

The investment arm of Canada-based Fairfax Financial Holdings Ltd. increased its stake in Eurobank to above 20% from just over 15%, the bank said in early April. Fairfax wasn't immediately available to comment on whether it has bought or sold shares more recently.

In a letter to shareholders in March, Fairfax Chairman and Chief Executive Prem Watsa said, "while our total Greek investments are below cost today, we expect them to do well over the long term."

Wellington Management Group LLC, a Boston-based fund manager, held 103.5 million shares in Piraeus, the bank said in March, a holding that was worth around €50 million as of Friday.

Separately, Wellington cut its Alpha Bank stake slightly, the bank said in April, but it still held 208.6 million shares that are currently worth around €70 million.

Wellington declined to comment on its holdings or whether it has bought or sold shares since then.

Egon Vavrek, a senior emerging-market portfolio manager at APG Asset Management, which as of the end of 2014 owned €61 million worth of shares in Greece's four biggest banks, said he would consider increasing his exposure if substantial progress is made on resolving the country's debt crisis.

Mr. Paulson said recently that it would be a mistake for Greece to default. "It would be disastrous for Greece's economy and citizens," he said.

"A default would be destabilizing to the European financial system, but manageable," he added.

Paulson increased its stake in Alpha Bank to 208.4 million shares, Alpha said in May last year, a holding worth more than €140 million a year ago that now has a value of around €70 million. Paulson & Co. declined to comment on any of its Greek holdings.

Greece's government aims to have an agreement with its international creditors by Sunday, but European officials warn that the country is still far from offering sufficient economic overhauls to unlock bailout funds.

Eleni Papoula, an analyst at Berenberg, said that while the majority of investors "seem to be taking a wait-and-see approach until a deal is confirmed" and ahead of major scheduled debt-repayment deadlines over the coming weeks, "valuations make the Greek bank stocks look very attractive to some."

Three of the four biggest Greek banks are trading higher on the week. But strategists and investors warned that liquidity is incredibly thin, accentuating any moves.

"Some days trading in Greek stocks and bonds is so thin that the moves we see on the screen mean nothing," said Julian Jacobson, a partner at privately owned investment manager FPP Asset Management LLP. FPP has small holdings in Greek banks and has in the past owned Greek sovereign bonds.

Write to Josie Cox at josie.cox@wsj.com and Rob Copeland at rob.copeland@wsj.com

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