(Updates yields, adds analyst quotes)

By Kate Duguid

NEW YORK, Oct 1 (Reuters) - U.S. Treasury yields on Tuesday afternoon hovered near session lows hit after the Institute for Supply Management (ISM) reported its U.S. manufacturing activity index fell in September to its lowest level in a decade.

The ISM said in the morning that its index of national factory activity fell to 47.8, the lowest reading since June 2009, as trade tensions between China and the United States kept straining business conditions. A reading below 50 signals the domestic factory sector is contracting.

"This raises the prospects of prolonged sub-par economic growth. You're not in recession territory, but you're definitely headed in that direction," said Stan Shipley, macro research analyst at Evercore ISI.

The two-year yield fell to a three-week low, last down 6.4 basis points to 1.558%. The two-year yield is a proxy for investor expectations of Federal Reserve interest-rate policy. Forecasts that the Fed will cut rates at its October meeting rose to 62.5% on Tuesday from 39.6% the previous day according to CME Group's FedWatch tool.

The benchmark 10-year yield was down 2.2 basis points to 1.649%, falling slower than the two-year yield and as a consequence, steepening the yield curve to 9.1 basis points, the highest in two weeks. The spread between the two- and 10-year yields was 4.1 basis points at Monday's close.

Investors will now focus on labor market data with the ADP private sector payrolls report due on Wednesday and the government's nonfarm payrolls report on Friday.

"If broader cracks in the labor market begin to emerge, look for (10-year yields) to retest 1.50% in short order with the 12-month low of 1.43% a subsequent target if the rally extends beyond there," wrote Ian Lyngen, head of U.S. rates strategy at BMO Capital Markets.

Treasury yields had been elevated prior to the manufacturing data. They were driven higher after Japanese government bond (JGB) prices slumped across the board on Tuesday after a 10-year debt auction received weak investor demand, with Japan's key 10-year futures contract posting the biggest daily fall in three years.

"We dissected the reasons why the sovereign yield auction in Japan was so poor. And the story you have there could happen here too. Low yields are not attractive for investors. This is going to be an increasingly important issue here as our deficits continue to go higher and we need bigger and bigger auctions," said Shipley. (Reporting by Kate Duguid; Editing by Catherine Evans, Nick Zieminski and David Gregorio)