* Euro retraces gains after failure to rise above $1.33 * 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 slipped to a near three-week low against the dollar on Thursday as optimism over a U.S. budget deal quickly faded and investors looked ahead to the release Friday of a closely watched U.S. government jobs report. The euro had surged early on Wednesday after U.S. lawmakers reached an agreement to avert a "fiscal cliff" of austerity measures, in the way of huge tax hikes and spending cuts, that some economists feared would tip the world's biggest economy back into recession. But the euro surrendered those gains as investors turned their focus to further 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. The euro was last down 0.6 percent to $1.3109, in a second straight session of declines. It had earlier dropped to $1.3082 according to Reuters data, the weakest since Dec. 14, after stop-loss sell orders were triggered below $1.3090. Traders said the euro's failure to break above $1.33 in the previous session drew sellers into the market. Strategists said the current 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. NONFARM PAYROLLS 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. Later in the session, the Federal Reserve will release the minutes of its most recent policy meeting. On Friday, the U.S. government will release its closely watched 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. The euro fell 1.1 percent to 113.89 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 0.5 percent to 86.89 yen, after climbing as high as 87.36 earlier in the global session, the highest since July 2010. Over the past few weeks, the yen has weakened on expectations that a new Japanese government led by Prime Minister Shinzo Abe will push the Bank of Japan into further monetary easing to beat deflation. Analysts said the yen is likely to remain vulnerable until the BOJ's policy meeting on Jan. 21-22. The Australian dollar rose 0.1 percent to $1.0508 after a 1-percent rally the previous day, supported by data showing China's services sector expanded in December.