By Min Zeng

Investors sought safety in U.S. Treasury bonds on Tuesday, driven by concerns over the potential for Greece defaulting on its debt.

In recent trading, the yield on the benchmark 10-year Treasury note was 2.186%, compared with 2.211% on Friday, according to Tradeweb.

Bond prices rise as their yields fall. The bond market was shut Monday for the Memorial Day holiday.

A string of U.S. data kept the bond market's price gain in check. Durable goods for April fell by a larger-than-forecast 0.5%, but details offered some positive sign on business investments. New home sales increased and a monthly gauge of consumer sentiment toward the growth outlook brightened.

A $26 billion sale of two-year Treasury notes is due at 1 p.m. Tuesday, followed by a $35 billion sale of five-year notes on Wednesday and a $29 billion sale of seven-year notes on Thursday.

For months, Greece's government and its international creditors have been stuck in drawn-out negotiations over terms for funding. The deadline for cash-strapped Greece to repay loans to the International Monetary Fund is due early next month.

Reflecting the worries, Greek government bonds sold off Tuesday, sending the yield on the 10-year Greek bond higher by over 0.4 percentage point to 11.9%.

"Greece is quickly approaching the day of reckoning," said Larry Milstein, head of government and agency trading at R.W. Pressprich & Co. in New York. "If Greece leaves the eurozone, it could roil the market for a time and lead to a safety bid to Treasury bonds."

Contagion has been contained over the past few months. Some investors still expect a deal to be reached to avoid a default. Meanwhile, the European Central Bank has been buying bonds since March to support the economy and investors say this monetary backdrop would help contain the risk of spillover.

Tuesday, the yield on the 10-year government bond in Spain rose by 0.1 percentage point to 1.881%. The yield remains near a record low.

Still, some investors are prepared for a worse outcome.

Mark Dowding, co-head of investment-grade debt at BlueBay Asset Management, which oversees $62.9 billion of assets, said he has reduced holdings of eurozone's government bonds anticipating a rise in volatility.

"The risk of a default occurring in the next month is as high as 50%," said Mr. Dowding. "Were this to occur, we expect there to be meaningful volatility in markets as investors worry about the stability of monetary union."

Another big focus for bond investors has been when the Federal Reserve will start raising short-term interest rates.

A major shift by the central bank into a tightening mode for the first time since 2006 would send prices of outstanding bonds lower as higher policy rates make newly sold bonds more attractive to buy.

Bond investors have benefited from a strong price rally in the bond market since the start of 2014. But the recent bond-market rout underscores that with bond yields near historical lows, even a moderate rise in yields would chip away the slim interest payments and inflict pain on bondholders.

The 10-year Treasury yield increased by 0.07 percentage point for the week. The yield touched 2.366% earlier in May, the highest intraday level since November. It was 2.173% at the end of 2014.

Federal Reserve Chairwoman Janet Yellen reiterated Friday that the central bank is on track to raise interest rates sometime this year. Ms. Yellen continued to expect the U.S. economy to rebound from a soft patch during the first quarter.

Mixed economic releases over the past few weeks have raised the question over how robustly the U.S. economy is rebounding. Many investors expect the Fed to wait until late this year to raise interest rates.

Some traders caution that the Fed may raise rates sooner than many investors expect, a case that would rattle the bond market and send yields climbing.

Write to Min Zeng at min.zeng@wsj.com