* Yields at 3-week highs before supply * Treasury to sell $32 bln in three-year notes * Four-week bill auction to see strong demand as supply dwindles * Fed to buy $1.25 bln to $1.75 bln in bonds due 2036-2043 By Karen Brettell NEW YORK, May 7 U.S. Treasuries yields rose to new three-week highs on Tuesday as traders prepared for the sale of $32 billion in new three-year notes, the first of $72 billion in new coupon-bearing bond sales this week. U.S. government bond yields have surged since Friday's better than expected jobs report for April as traders had positioned for a much gloomier number. At the same time, many see yields as unlikely to march significantly higher from here unless there are new signs that the economic recovery is not slowing as much as feared. "One decent number is not strong enough to completely change the mood of market players," said Jason Rogan, managing director of Treasuries trading at Guggenheim Partners in New York. "We're getting close to a point where you might start to see some buying." Benchmark 10-year notes yields rose to 1.78 percent on Tuesday, up from 1.76 percent on Monday and the highest since April 12. Rogan sees levels of 1.80 percent to 1.82 percent as likely to attract new buying from fund managers or central banks. Thirty-year bonds yields inched up to touch the key 3 percent level, up from 2.98 percent on Monday and also the highest since April 12. The higher yields may help demand for this week's new supply, with the Treasury also due to sell $24 billion in 10-year notes on Wednesday and $16 billion in 30-year bonds on Thursday. The Treasury will also auction $20 billion in four-week Treasury bills on Tuesday, the lowest amount for that auction since an auction held in 2001, which is likely to be snapped up by investors hungry for the collateral. The Treasury has been slashing its supply of short-term debt as it accumulates more cash from tax receipts, and as the government prepares for a new round of wrangling over the U.S. debt ceiling later this month. Returns on one-month bills have plunged on the declining supply, with one-month bill yields dropping to only half a basis point on Tuesday, down from 3 basis points last week and from over 10 basis points in late February. The Federal Reserve will also buy between $1.25 billion and $1.75 billion in bonds due from 2036 to 2043 on Tuesday as part of its ongoing bond purchase program.