* U.S. Treasury sells $32 bln 3-year notes, high yield 0.342 pct * 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 Luciana Lopez NEW YORK, April 9 (Reuters) - Prices for longer-dated U.S. Treasuries slid on Tuesday as investors extended a selloff after last week's rally and before debt auctions later in the week, but an undercurrent of worries about the global economy tempered losses. An auction of $32 billion in 3-year debt by the U.S. Treasury proved soft on Tuesday, with a weak bid-to-cover ratio and weak indirect takedown. "The small buyside takedown suggests to us that demand for the 3-year note is not all that huge at the moment," said Thomas Simons, a money market economist at Jeffries & Co. Thirty-year bonds fell 13/32 in price to yield 2.931 percent after the auction. Those bonds saw a sharp rally last week as data pointed to continued struggles in the U.S. economy, culminating with sharply disappointing jobs data on Friday that saw benchmark yields touch their lowest so far this year. That run-up left prices too high for some, said Kim Rupert, managing director of global fixed income analysis at Action Economics LLC in San Francisco. That could dampen demand at an auction of $13 billion in 30-year bonds on Thursday. If yields back up, "we'll see some guys step in for a trade," Rupert said. "But if it doesn't cheapen up appreciably and it's still trading where it is today, I think the auction goes again pretty soft." Ten-year notes traded near flat ahead of Wednesday's auction of $21 billion in 10-year notes. Those yields hovered at their short-term chart supports after they failed to climb back to levels before last Friday's release of the government's dismal payrolls report. Losses in Treasuries were limited by persistent worries not just about the U.S. economy, but also the euro zone sovereign debt crisis and bellicose rhetoric from North Korea. "It's a real wild card," Rupert said. Also supporting Treasuries, the Federal reserve bought $1.57 billion in government bonds that mature in Feb. 2037 to Feb. 2043, part of efforts to buoy the economy and reduce unemployment. "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 remained supported as well 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. Still, investors paused last week's buying spree as 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 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 negative for 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 Bank of Japan'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. The spread was last quoted at minus 6.00.