* Euro falls after Fed minutes show wariness on bond buying * Focus shifts to Friday's jobs data, debt ceiling debate * Yen up vs dollar after hitting highest since July 2010 NEW YORK, Jan 3 (Reuters) - The euro fell to a three-week low against the dollar on Thursday after minutes from the Federal Reserve's latest meeting indicated that while the U.S. central bank looks set to continue buying bonds, some policymakers believe it will be appropriate to "slow or stop asset buys well before the end of 2013." The minutes from the Fed's December meeting showed a growing reticence about further increases in the central bank's $2.9 trillion balance sheet, which it expanded sharply in response to the financial crisis and recession of 2007-2009. The euro was already under pressure as optimism over a U.S. budget deal quickly faded and investors looked ahead to the release Friday of the closely watched U.S. government monthly jobs report. The Fed minutes only accelerated the move. "The minutes of the Federal Reserve's December FOMC monetary policy meeting revealed a somewhat surprising level of concern among the ranks of central bankers regarding the long-term impact of the bank's asset purchase program, or quantitative easing," said Omer Esiner, chief market analyst, at Commonwealth Foreign Exchange in Washington. The euro was last down 1 percent at $1.3051, in a second straight session of declines. The session low of $1.3047 was the weakest since Dec. 13, according to Reuters data. Traders said the euro's failure to break above $1.33 on Wednesday drew sellers into the market early in the day. Some $4.186 billion in euros changed hands on Thursday, according to Reuters Dealing, compared with $6.076 billion the first Thursday of December. The euro had surged early on Wednesday after U.S. lawmakers reached an agreement to avert a "fiscal cliff" of austerity measures that included huge tax hikes and spending cuts, which some economists feared would tip the world's biggest economy into recession. But the euro surrendered gains as investors turned their focus to the budget battles ahead. Analysts say the market could be set up for volatility as President Barack Obama and congressional Republicans tussle over the next two months. "After yesterday's big move, the market is just taking a little bit of a breather here," said Amarjit Sahota, director of Klarity FX in San Francisco. "Focus is now turning to the debt ceiling and the spending cuts, which still need to be agreed." Republicans, angry that the fiscal cliff deal did little to curb the federal deficit, promised to use the debt-ceiling debate to win deep spending cuts next time. Strategists said the weakness in the euro could persist as the euro zone economy falls deeper into recession and on increasing prospects of an interest rate cut by the European Central Bank. Highlighting market concerns that the U.S. deficit issue remain unresolved, ratings agency Moody's Investors Service said the United States must do more to rescue its Aaa debt rating from its current negative outlook. Standard & Poor's said the fiscal deal does not affect its negative view of the U.S. credit outlook, and said more work remained ahead for policymakers. EURO SAGS VS YEN Adding to gains in the dollar versus the euro was data showing U.S. private-sector employers added more new jobs than expected last month. Separate data showed U.S. initial jobless claims rose last week, but the trend remained consistent with steady job growth. On Friday, the U.S. government will release its monthly nonfarm payrolls report. The economy likely added 150,000 in December, according to a Reuters survey of economists, up from 146,000 in November. The unemployment rate is expected to hold steady at 7.7 percent. Against the yen, the euro fell 1.1 percent to 113.84 yen as investors bet that its rise to an 18-month high of 115.99 yen on Wednesday was too far, too fast. "Euro/yen at around 115 levels was starting to look a bit overdone and the euro may actually lose ground against the yen in the coming weeks," said Colin Asher, senior economist at Mizuho Corporate Bank. The dollar fell 0.1 percent to 87.26 yen, after climbing as high as 87.36 earlier in the global session, the highest since July 2010.