U.S. Treasury were higher after reports surfaced that Greece may suspend referendum if debt talks resume.
Yields hit session lows earlier after an euro zone official told Reuters there was "no way" the Eurogroup will release cash for Greece to meet its debt repayment to the International Monetary Fund.
The debt-laden European country is likely to miss Tuesday's deadline to repay $1.8 billion to the International Monetary Fund (IMF), while the terms of its existing bailout terms expire at midnight.
Safe-haven bonds in the U.S. and Germany have been given a boost from growing concerns that Greece is moving closer to a "disorderly" exit from the euro zone.
Greece has shut banks, imposed capital controls and called a referendum on Sunday on austerity cuts in the aid program proposed by creditors.
In addition to Greek jitters, safe-haven bonds could draws support from concerns about a possible debt default in Puerto Rico, which analysts say could have ramifications for emerging markets.
Puerto Rico Gov. Alejandro Garcia Padilla late on Monday said the island,whose economy is closely tied to the U.S., needs a restructuring plan for its $72 billion in debt.
In U.S. data, the Conference Board, an industry group, said its index of consumer attitudes rose to 101.4, matching the level set in March and higher than a downwardly revised 94.3 in May. Economists had forecast a June reading of 97.3, according to a Reuters poll.
Separately, U.S. single-family home prices rose in April from a year earlier, but at a slower pace than forecast, a closely watched survey said on Tuesday.
The data highlight of the week is the U.S. non-farm payrolls report, which will be published on Thursday.
—Reuters contributed to this report.