Long-dated U.S. Treasury yields hovered close to multidecade-high levels in Asian trade on Tuesday, as investors remained concerned over fiscal, inflation and debt sustainability risks.

The 10-year Treasury yield was down 0.8 basis points at 5.301%, while the 30-year yield was steady at 5.663%, according to Tradeweb. They reached levels of 5.349% and 5.703% on Monday, respectively, levels unseen since 2002.

Attention will turn to the Treasury's $58 billion auction of three-year notes on Tuesday.

With levels attractively high for buyers at the short end of the curve, and with odds of a Federal Reserve rate hike in October declining, particularly after Friday's weak U.S. jobs data, the auction is expected to go smoothly.

"Focus is likely to move to the three-year auction tonight from the U.S. Treasury," said Jens Peter Sorensen, chief analyst at Danske Bank. "The outright level of close to 5% is looking attractive despite higher-than-expected ISM service prices released yesterday," he said in a note.

Investors priced a 23% probability of a quarter-point Fed rate increase on Oct. 28, sharply down from as high as 70% at the start of last week, although at least three rate hikes are priced over the coming year, LSEG data showed.