* U.S. Treasury to sell $32 bln 3-year note supply * U.S. Fed buys $1.57 bln in long-dated bonds * Bullard says Fed still has room to buy more bonds * Thirty-year swap spread widens, approaches zero By Richard Leong NEW YORK, April 9 (Reuters) - U.S. government debt prices edged higher on Tuesday in advance of a $32 billion auction of a new three-year note issue, which was the first part of the $66 billion in total supply of longer-dated debt this week. Benchmark yields held at their short-term chart supports after they failed to climb back to levels prior to the release of the government's dismal payrolls report last Friday. They fell to session lows on buying tied to the Federal Reserve's regular Treasuries purchases aimed to reduce unemployment. The Fed earlier bought $1.57 billion in government bonds that mature in Feb. 2037 to Feb. 2043. "The market is not going to go down a lot because of the Fed buyback," said Aaron Kohli, an interest rate strategist at BNP Paribas in New York. The U.S. bond market also remained supported by bets in anticipation of heavy purchases from Japanese insurers and pension funds as they seek higher-yielding debt overseas after the Bank of Japan's plan to double its monthly asset purchases in a bid to stimulate its sluggish economy. Last week's batch of weaker-than-expected data on jobs and business activity added safehaven bids for Treasuries, whose yields fell to their levels lowest of the year on Friday. Still the bond market rally has paused on the perception that it was overdone for now and the expected buying from Japan has not materialized, analysts and traders said. "We ran ahead too far too soon and we still have supply to deal with," said Thomas Roth, executive director of U.S. government bond trading at Mitsubishi UFJ Securities USA in New York. In "when-issued" activity, traders expected the upcoming three-year note issue to yield 0.3420 percent, lower than the 0.411 percent yield at last month's three-year auction. After the three-year note sale at 1 p.m. (1700 GMT), the U.S. Treasury Department $21 billion in 10-year notes on Wednesday and $13 billion in 30-year bonds on Thursday . On the open market, benchmark 10-year Treasury notes were up 4/32 in price at 102-13/32, yielding 1.731 percent, down 1.4 basis point from late on Monday. Bond prices bounced up from their earlier lows, partly on comments from St. Louis Federal Reserve President James Bullard who told CNBC television that the U.S. central bank still has more room to buy bonds to help support the economic recovery. The 10-year yield briefly broke above its 200-day moving average of 1.7431 percent, according to Reuters data. It also retraced 50 percent of last week's decline due to buying spurred by Bank of Japan's $1.4 trillion stimulus plan and the payroll report that showed a stunningly weak jobs gain of 88,000 in March. In the derivatives market, the spread on U.S. 30-year interest swap rates over 30-year Treasury bond yields crept closer to zero earlier. This measure of the difference between long-term U.S. government and private borrowing costs has been stuck in negative territory more than four years on the view the long-term cost for U.S. government debt will remain higher than that of the private sector due to rising costs for its social programs and erosion of the dollar's value against other major currencies. This widening of the 30-year spread has been fueled by speculation that some dealers must close out 30-year swaps, which they use to hedge investments known as power reverse dual currency notes (PRDC) they sold to investors. The yen's rise against the U.S. dollar led dealers to hold 30-year swap positions to ensure cash flows to pay PRDC investors, but the recent rapid weakening of the yen due to the BoJ's aggressive easing has spurred bets dealers will exit these long-dated swaps. The 30-year swap spread tightened to minus 3.00 to minus 3.50 basis points when the yen flirted with a four-year low against the dollar within striking distance of the 100 yen threshold during Asian trading. It was last quoted at minus 4.50 to minus 5.00 basis points, which was 0.75 basis points tighter than Monday, when the yen gained versus the dollar.