The growing threat of Greece's exit from the eurozone rattled financial markets Monday, as investors dumped stocks world-wide and moved to the safety of U.S. and German government bonds.

The Dow Jones Industrial Average fell more than 350 points, or 2%, its biggest percentage decline since October, putting the blue-chip index into negative territory for the year. Stock markets across Europe dropped, many suffering their worst routs in years. Italian stocks fell more than 5%, and Germany's DAX index slid 3.6%.

The yield on the benchmark 10-year Treasury note tumbled to its biggest one-day decline since Nov. 1, 2011. Yields fall when prices rise.

With Greece's stock market closed, some investors who wished to make bets on the country's markets turned to the Global X FTSE Greece 20 ETF. The small exchange-traded fund, which tracks Greek stocks, fell 19% Monday.

The sudden breakdown in talks between Greece and European officials forced investors to confront the possibility Greece would default and leave the eurozone.

Many investors said they believed a Greek exit would have limited impact on financial markets. But they said they were wary of holding too many risky assets ahead of what is likely to be a turbulent week leading up to a planned Greek referendum Sunday.

China has also been a cause of concern for many investors. Stocks there Monday extended a sharp two-week-long slide, despite a surprise interest-rate cut. Adding to worries, Puerto Rico said it would be unable to pay all its debts.

On Thursday, investors will focus on the U.S. Labor Department's employment report for June.

"People are getting nervous, as what seems like—all of a sudden, out of the blue—you have multiple hot spots," said Ian Winer, director of equity trading at Wedbush Securities. "They're just trying to protect their gains on the year and position themselves in the event that this becomes a more dramatic selloff."

Investors are mindful that stocks are in their seventh year of a bull run, trading near records.

In Europe, markets have rallied sharply in recent months on signs of improved growth and helped by stimulus from the European Central Bank.

The Dow fell 350.33 points Monday to 17596.35. Still, the index remains just 3.9% below its all-time closing high.

The 10-year U.S. Treasury yield ended Monday at 2.333%, from 2.480% Friday.

The euro at first sank against the dollar Monday, but then gained as some investors bet the latest flare-up in the Greek crisis would prompt the U.S. Federal Reserve to push back a decision to raise interest rates—potentially into 2016. Investors in past months have flocked to the dollar in anticipation of higher rates that would bolster returns, but the rally has lost steam amid patchy U.S. economic performance.

Gyrations in currency markets prompted the Swiss National Bank to intervene to tamp down the value of the franc against the euro, the central bank's chairman, Thomas Jordan, said Monday.

Despite the stock-price declines, traders said volumes were only slightly higher than in past trading sessions.

"People are not panicking on this. It's not a fire sale," said Ryan Larson, head of U.S. equity trading for RBC Global Asset Management. "For the most part, the market is taking this fairly in stride."

Traders and analysts said stock markets are likely to hold up better than they did in 2011 and 2012, when anxiety ahead of Greece's bailout snowballed into angst about the financial stability of bigger European economies, such as Spain and Italy.

From early July to late September of 2011, the Stoxx Europe 600 index plummeted more than 20%. The index Monday ended down 2.7% and is off 6.7% from its record reached just two months ago.

Bond yields on Greek debt surged Monday, but the rise was more modest for bonds from Italy, Spain and Portugal—highly indebted countries that in past years were seen as vulnerable to spillover from Greece. Yields on those countries' debt remain near historic lows.

"Fundamentally, things are better now. There's much more central-bank support. The stability of the world is in a better place than it was three or four years ago," said Scott Wren, senior global equity strategist at Wells Fargo Investment Institute. "The 'contagion' fear to the same degree is not there."

Little of Greece's government debt is still held by banks and private investors outside the country, limiting the threat that heavy losses could ripple around the currency bloc.

And the European Central Bank's bond-buying program, launched in March, provides a backstop for government debt markets.

"It's a pretty scary thing for Greece, what could happen if they were to drop out of the eurozone. But not necessarily for the rest of the eurozone, or for financial systems in the rest of the world," said Mr. Wren.

Some investors, however, said the current low bond yields are unsustainable given the turmoil in Greece.

"If we get a Greek exit, you have to say the potential for other countries to exit the eurozone is suddenly no longer negligible. That would need to be reflected in bond prices," said Mark Dowding, co-head of investment-grade fixed income at BlueBay Asset Management.

A number of investors saw the selloff as a good chance to buy European stocks, reasoning that the turmoil in Greece is unlikely to hold back a broader economic recovery in the region.

"Over our six-month investment horizon, ECB action should mitigate contagion effects to other markets or economies in case of a Greek exit," said Mark Haefele, chief investment officer at UBS Wealth Management.

Much uncertainty remains. If Greek voters reject the creditors' demands in Sunday's referendum, investors' faith that Greece won't trigger a financial crisis will be tested anew.

In that scenario, stock markets in the euro area could fall by about 15% on "uncertainty and fear," said Eric Chaney, head of investment strategy at AXA Investment Managers, which oversees €689 billion of assets.

A reaction Monday that some might see as mild means there is room for further declines, said Ewout van Schaick, who oversees more than €20 billion ($22.6 billion) at NN Investment Partners. Mr. van Schaick said he is considering selling some of his European stockholdings to protect his portfolio from short-term losses.

"It's going to be a volatile week in the market," he said.

Write to Tommy Stubbington at tommy.stubbington@wsj.com, Corrie Driebusch at corrie.driebusch@wsj.com and Josie Cox at josie.cox@wsj.com

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