* Private index on euro zone business hits 10-month high * Fed to buy $1.25 bln to $1.75 bln debt due 2036-2042 * ISM U.S. services activity data on tap * U.S. to sell a record amount of one-month bills By Richard Leong NEW YORK, Feb 5 (Reuters) - U.S. Treasuries prices fell on Tuesday after less worrisome data on European business activity in 10 months raised hopes of an economic recovery in the region, stoking selling of safe-haven government debt. The market sell-off was also spurred by lower yields on Italian and Spanish sovereign debt after they jumped on Monday on worries about possible major political shake-ups in euro zone's third and fourth biggest economies. Another factor exacerbating the weakness in Treasuries prices was a revived reduction of long-dated hedges on debt securities known as power reverse dual currency notes. Long-dated yields have risen recently as dealers have been unwinding hedges on these "exotic" products due to weakness in the Japanese yen against the dollar, traders and analysts said. "The key in the near term is the movement in Spanish and Italian yields. We are seeing a small reversal after Monday's move. We are seeing Treasuries yields rising in response," said Brian Reynolds, chief market strategist at Rosenblatt Securities in New York. Benchmark 10-year Treasuries notes were 15/32 in price at 96-19/32, yielding 2.011 percent, up 5.4 basis points from late on Monday when the 10-year yield hit 2.059 percent, the highest level in more than nine months. The 30-year bond were 27/32 lower in price at 91-8/32 for a yield of 3.208 percent, up 4.6 basis points from Monday's close. The weakness in Treasuries stemmed from a partial recovery in Italian and Spanish government debt when investors sold them on corruption allegations of Spanish Prime Minister Mariano Rajoy's government and growing chances that former Italian prime minister Silvio Berlusconi might regain office. The yield on 10-year Spanish government notes fell from a six-week high to 5.37 percent, down 6 basis points from Monday's close, while the yield on Italian sovereign debt slipped to 4.43 percent, down 4 basis points from late on Monday. Concerns about euro zone's political developments and their implications on the region's handling of its fiscal crisis coincided with some signs that the euro zone's economy may be turning the corner. Earlier Tuesday, Markit's Eurozone Composite PMI, based on business activity across thousands of companies, and a good gauge of economic growth, rose in January to a 10-month high of 48.6 from 47.2 in December - an improvement on the preliminary reading of 48.2. The Institute for Supply Management will release its January figures on the U.S. services sector at 10 a.m. (1500 GMT). Economists forecast the ISM non-manufacturing index likely decline to 55.2 points in January from 55.7 in December. On the supply front, the U.S. Federal Reserve will buy $1.25 billion to $1.75 billion in Treasuries that mature between Feb. 2036 and Nov. 2042 at 11 a.m. (1600 GMT), which is part of its $44 billion purchase of Treasuries in February. The U.S. Treasury Department will sell at 11:30 a.m. (1630 GMT) $45 billion in one-month bills, which is the a record amount for this maturity.