Treasury Department auctions $24 billion of 3-year notes at a high yield of 1.125%

Bond yields held gains on Tuesday after the U.S. government's auction of three-year Treasury notes, the first batch of this week's $58 billion offering of new debt supply.

The Treasury Department auctioned $24 billion in three-year notes at a high yield of 1.125 percent, the highest since April 2011. The bid-to-cover ratio, an indicator of demand, was 3.33, little changed from 3.34 in May.

Indirect bidders, which include major central banks, were awarded 50.7 percent, while direct bidders, which includes domestic money managers, brought 9.7 percent.

"The 3 yr note auction, very sensitive to expectations of Fed rate hikes, was uneventful," said Peter Boockvar, chief market analyst at the Lindsey Group. "Bottom line, the highest yield in 3 months led to only ordinary demand as the complexion and trend of the U.S. Treasury market has clearly changed over the past 2 months."

Three-year note yields were trading at 1.09 percent, up from 1.059 percent on Monday, while 2-year note yields traded up 2 basis points at 0.70 percent.

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Ten-year bond yields hit a eight-month high earlier as upbeat economic reports renewed expectations for a rate rise this year.

The Job Openings and Labor Turnover (JOLT) survey showed U.S. job openings rose to 5.4 million in April—their highest since December 2000.

Separately, wholesale Inventories rose 0.4 percent in April, versus forecasts for a 0.2 percent increase.

The yield on the 10-year note hit a session high of 2.44 percent—the highest since October 3 2014. It was last up 4 basis points at 2.42 percent. Meanwhile,

Concerns about Greece and its ability to avoid a debt default have renewed demand for safe-haven bonds after a hefty sell-off last week that sent benchmark yields in the U.S. and Europe to their highest levels this year.

Symbol
Price
 
Change
%Change
US 3-MO
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US 1-YR
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US 2-YR
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US 5-YR
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US 10-YR
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US 30-YR
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The Wall Street Journal reported on Tuesday that Greece and its international creditors were discussing an extension of the cash-strapped country's bailout program through to March 2016.

Read MoreGreece could get bailout extension to March 2016: Report

Elsewhere, Germany, which has Europe's biggest bond market, said it has started marketing a 30-year inflation-linked euro bond.

U.S. stock were trading slightly lower. The dollar index, which measures the value of the dollar against other major currencies, stood at 95.36 – little changed on the day.

CNBC's Dhara Ranasinghe contributed to this report.