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US Treasurys gain on Greek woes

U.S. Treasurys were on a firm footing on Tuesday as worries about Greece bolstered the market's safe-haven appeal ahead of a U.S. Federal Reserve meeting.

The benchmark 10-year Treasury yield, which moves in the opposite direction to the price, was trading at 2.3220 percent —down from 2.36 percent in late New York trade Monday and 17 basis points below an eight-month peak hit last week.

U.S. Markets Overview: Treasurys chart

Concern that Greece is heading towards a default of its debt that could end with the country leaving the euro zone has boosted demand for bonds, which have seen heightened volatility in recent months as investors reassess the outlook for inflation and monetary policy.

Read MoreGreece weighs while traders await Fed

"It would create a great deal of dislocation in the short-run if Greece left the euro zone. It's not an eventuality we would like to see, but at the end of the day Greece is a relatively small economy," John Buckingham, chief investment officer of AFAM Capital, told "Squawk Box" Asia.

The Federal Reserve, meanwhile, is not expected to announce any changes to its monetary policy when it concludes a two-day meeting on Wednesday.

Still, the Fed's post-meeting statement will be scrutinized amid talk that the Fed could lift interest rates in September for the first time since 2006 amid improving economic conditions.

Read MoreGreece defiant, accused of 'breaking all the rules'

On the data front, U.S. housing starts fell in May after a hefty increase the prior month, but a surge in permits for future construction to a near eight-year high suggested the pullback was temporary and pointed to underlying strength in housing.

Groundbreaking dropped 11.1 percent to a seasonally adjusted annual pace of 1.04 mullion units, the Commerce Department said on Tuesday. That partially reversed April's large gain. April starts were revised up to a 1.17 million-unit rate, the highest since November 2007.

Economists polled by Reuters had forecast housing starts falling to a 1.10 million-unit pace last month after a previously reported 1.14 million-unit rate.

That follows Monday's National Association of Home Builders sentiment survey, which showed confidence rising to its highest level in nine months.

Elsewhere, U.S. stock futures pointed to a weak start for Wall Street shares amid jitters over Greece.

—Reuters contributed to this report.