* Treasury auctions $32 billion in three-year notes * After selloff, higher yields attract buying * Fed buys $3.71 billion in notes due 2018-2019 By Karen Brettell NEW YORK, Jan 8 (Reuters) - Prices for U.S. Treasuries advanced on Tuesday as higher yields drew in buyers and the year's first sale of coupon-bearing Treasury debt saw strong non-dealer bidding. The Treasury sold $32 billion of three-year notes on Tuesday at a high yield of 0.385 percent, just about where the market had expected. The high direct takedown in this and the previous three-year auction could signal "a shift in investor bidding patterns at auctions, where buyers bypass dealers and go straight to the Treasury, while still able to clear the auction near the WI (when issued) levels," wrote Nomura analysts after the sale. Treasuries held gains after the sale. U.S. government debt sold off sharply last week after lawmakers agreed to a deal to avoid a package of automatic tax hikes and spending cuts, the so-called fiscal cliff, that could have pushed the world's biggest economy back into recession. Investors are also weighing whether the Fed might end its bond purchase program before year-end, after minutes from the Fed's December policy meeting released last week showed growing unease on continuing the buybacks. The higher yields are likely to help demand for $66 billion in new debt sales this week. "The recent back-up creates a buying opportunity," said Sean Simko, portfolio manager at SEI Investments in Oaks, Pennsylvania. The Treasury will sell $21 billion in 10-year notes on Wednesday and $13 billion in 30-year bonds on Thursday. Benchmark 10-year Treasures were last up 10/32 in price to yield 1.866 percent, down from 1.90 percent late on Monday but up from around 1.70 percent at year-end. Thirty-year bonds gained 22/32 in price to yield 3.065 percent, down from 3.10 percent on Monday, and up from 2.87 percent at year-end. Market talk that France's sovereign debt rating was about to be downgraded also lifted Treasuries' safe-haven appeal. A senior French official told Reuters the rumor was "erroneous." "The initial jump was on the rumor of the French downgrade, but now there is decent buying going on," said Charles Comiskey, head of Treasuries trading at Bank of Nova Scotia in New York. With no large economic indicators on the horizon, Treasuries are expected to stay largely range-bound, albeit in a higher yield range after the past week's push. The market's next focus is likely to be a new round of wrangling in Washington where lawmakers face tough battles on how to cut spending and reduce the deficit, as the country also pushes up against its debt ceiling. "People are focusing back on the bigger picture, we still have the debt ceiling in front of us, we still have growth but albeit at a slower pace than everybody would like," said Simko. The Fed bought $3.71 billion in notes due 2018 and 2019 on Tuesday as part of its latest quantitative easing program, meant to stimulate hiring by reducing long-term borrowing rates.