TREASURIES-Bonds rise on stock losses ahead of supply

* Market advance as stock sell-off deepens

* Bonds' priciness curbs rise before $70 bln in supply

* Curve flattens in the wake of disappointing jobs data (Updates market action, adds quotes, changes byline)

By Richard Leong

NEW YORK, June 7 (Reuters) - The U.S. Treasuries market climbed on Monday, as struggling stocks and worries over Europe's fiscal crisis kept up a safety bid for bonds in advance of this week's $70 billion supply.

Wall Street struggled to rebound after Friday's sell-off due to a disappointing government report on payrolls. For latest market report, see [.N]

Concerns over deterioration in the labor market and the U.S. economic recovery make it difficult to convince anxious investors and bond dealers to part with safe-haven government bonds for the upcoming Treasury issues.

"People don't want to get too short ahead of the auctions," said Jeff Given, portfolio manager at MFC Global Investment Management in Boston.

The Treasury Department will sell $36 billion in three-year notes on Tuesday; $21 billion in 10-year debt on Wednesday and $13 billion in 30-year bonds on Thursday.

Analysts reckoned this week's coupon-bearing securities will see solid demand, as recent market volatility and economic uncertainties should support demand for Treasuries and other low-risk assets.

Treasuries have become expensive over the past two months even as bond bears have warned of the risk of owning U.S. government bonds, drawing the parallel of the fiscal predicament of Greece and other weaker euro zone nations.

"They are the highest quality asset and everyone still hates them," Richard Bernstiein, chief executive of his namesake investment firm, said of U.S. Treasuries at the Reuters Investment Outlook Summit in New York, (For other news from the Reuters Investment Outlook Summit, double-click on: here)

In the "when-issued" market, traders expected the upcoming three-year notes due June 2013 US3YTWI=TWEB to yield 1.1870 percent late Monday. This compared with the yield on the three-year issue US3YT=RR sold last month, which last traded at 1.152 percent, down from 1.161 percent late Friday.

The benchmark 10-year note US10YT=RR was up 12/32 at 102-27/32. Its yield which moves inversely to price was 3.17 percent, down 4 basis points on the day.

That yield level is a significant chart resistance for the 10-year note. If the market gains further, the 10-year yield will likely test the 3.10 percent mark, said William O'Donnell, head of U.S. Treasury Strategy at RBS Securities in Stamford, Connecticut.

Meanwhile, the two-year to 10-year segment of the yield curve narrowed on the day to 247 basis points, marking the yield curve's flattest late May, according to Reuters data.

The shrinking gap between short- and longer-dated yields signals traders bracing for slower growth and inflation.

That view was reinforced by Friday's weaker-than-expected 431,000 payroll increase in May, although some investors are confident the U.S. economy is decent shape.

"While the May data were a bit disappointing, the longer term trend still points to a gradual improvement in the broader labor picture in the United States," said Christopher Molumphy, chief investment officer for the Franklin Templeton fixed-income group in San Mateo, California.

(Additional reporting by Burton Frierson, Editing by Chizu Nomiyama)