* Strong 3-year notes sale fuels further gains
* Worries over Europe, fall in stocks revive safe-haven bid
* Rethink about fleeting U.S. jobs relief (Updates after auction, adds quote, latest prices)
By Burton Frierson
NEW YORK, Sept 7 Treasuries rebounded on Tuesday after three straight losing sessions, as waning confidence in the U.S. stock market and a strong auction of three-year notes gave the market a lift.
Investors were reassessing their initial relief about the latest U.S. monthly jobs report on Friday, which was less grim than expected. Resurgent worries over the euro zone's financial health also enhanced the allure of safe-haven bonds.
A well-received auction of $33 billion worth of three-year Treasuries, the first of this week's three bond offerings totaling $67 billion, added to the bond-positive sentiment.
The sale attracted bids worth 3.21 times the amount on offer, right on their six-month average. Yields at the auction came in slightly below expectations, indicating that investors were willing to pay a small premium to get hold of the bonds.
"There is a lot of cash in the system. There is a lot of demand at the front end," said Joe Larizza, director of governments and agencies trading in Vining Sparks in Memphis.
"You had a lot concerns about European stress tests, and that put a run back into the Treasuries market," Larizza said. "This sets up a positive tone for the rest of the week. We are looking at continued strength going into the auctions."
The benchmark 10-year note US10YT=RR was last up 21/32 in price, yielding 2.63 percent versus Friday's close of 2.70 percent.
A report on the European banking system reawakened fears about the region's financial health. A Wall Street Journal report raised questions about bank stress tests carried out in the euro zone earlier in the year.
Separately, Germany's banking association said the country's 10 biggest banks may need 105 billion euros ($141 billion) of additional capital under a revamp of banking rules designed to prevent future financial crises, which weighed on financial shares. For details see [ID:nLDE6850Q9] and [.N] (Additional reporting by Richard Leong and John Parry; Editing by Padraic Cassidy)