The yields on 10-year notes stood at 1.48 percent, compared with 1.47 percent in late U.S trading on Friday, when they fell to a record low of 1.44 percent.

That was the lowest level ever reached going back to the early 1800s, according to data gathered by Reuters, and sharply below the 2.40 percent in mid-March.

The yields on 30-year bonds stood at 2.54 percent, matching their level in late US trade on Friday, when they fell to a record low of 2.51 percent.

Friday's monthly US payrolls report showed employers added only 69,000 workers in May, far short of the 150,000 predicted by economists. The jobless rate unexpectedly edged up to 8.2 percent from 8.1 percent in April.

The dismal jobs report gave the US Federal Reserve more reason to extend its bond purchases after its "Operation Twist" stimulus program expires at the end of this month.

The Fed could even embark on a third round of quantitative easing, or QE3.

"Yields can't keep falling forever, but with bunds also at record lows, no one is ready to call the bottom yet," said a fixed-income fund manager at a European asset management firm in Tokyo.

Investors' flight to safety after the US data, as well as continuing fears about Spain's debt, pushed German 10-year yields to as low as 1.129 percent and German two-year yields further into negative territory On Friday.

Also underpinning demand for Treasuries on Friday, a private group's report showed slower US manufacturing growth in May.