Ten-year Treasuries held steady in price to yield 1.549 percent, down about 1 basis point from late US trade on Friday. The 10-year yield was not far from Friday's low of 1.537 percent, its lowest level since early June.

"The firmness reflects concerns about a worsening outlook for economic growth after the jobs data," said Tomohisa Fujiki, interest rate strategist for BNP Paribas Securities in Tokyo.

US employers hired at a dismal pace in June, with non-farm payrolls growing by just 80,000 jobs, the third straight month below 100,000.

The focus now is whether the 10-year yield will head toward its early June trough, Fujiki added. The 10-year yield had hit a low of 1.442 percent on June 1. That was the lowest level on records going back to the early 1800s, according to data gathered by Reuters.

Near-term event risks include a series of auctions of Treasuries this week, Fujiki said.

Treasuries showed little reaction to comments by Boston Fed President Eric Rosengren, who said on Monday that slow US economic growth will probably continue for quite some time as firms postpone hiring and investment in the face of an uncertain global economy.

Rosengren, a dovish policymaker who does not have a vote on the Fed's policy-setting committee this year, warned about the weak recovery in the US labour market.