* Euro zone govts disagree on whether to give Greece money * China trade data shows economy recovering * U.S. fiscal worries hit appetite for riskier assets By Julie Haviv and Gertrude Chavez-Dreyfuss NEW YORK, Nov 12 (Reuters) - The euro was unchanged against the dollar on Monday and nearing a recent two-month low in thin holiday trading, weighed down by uncertainty as to whether or not Greece will receive a new round of bailout money to help pay off its debt. The euro has dropped about 1.9 percent against the dollar so far in November and is seen having limited scope for gains even if Greece does receive aid. A bleak economic backdrop in the euro zone contrasts starkly with an improving U.S. economy and is viewed as a further headwind for the euro. Also favoring the safe-haven dollar against the euro is the a looming U.S. "fiscal cliff," a combination of big spending cuts and tax increases if Congress does not act to curb the budget deficit that some believe has the potential to send the economy into another recession. Trade in the United States was on the light side, with the government bond market closed in observance of the Veterans Day holiday. Greece stood at the forefront of investor concerns as euro zone governments disagreed on whether to disburse more money to the debt-ravaged country on Monday. Worries persisted even though the Greek government approved a tough 2013 budget, because of the lack so far of a consensus on how to make Greece's debts sustainable into the next decade. "Greece continues to be a thorn in the side of the European Union," said Neal Gilbert, currency strategist at GFT in Grand Rapids, Michigan. "Despite the Greek parliament passing an austerity-filled budget this weekend, investors are still concerned that Greece will not receive its next tranche of funds in time to avoid defaulting on its loans." Athens has to redeem 5 billion euros ($6.36 billion) worth of Treasury bills on Nov. 16 and had been counting on cash from the next tranche of aid to help cover that. Greece will get two more years to reach previously agreed budget goals, but the extra time will cost the euro zone an additional 32.6 billion euros, draft documents prepared for a meeting of euro zone finance ministers showed. The euro last traded flat at $1.2710, not far from a two-month low of $1.2688 touched on Friday, according to Reuters data. Traders cited stop-loss sell orders below $1.2685. Intraday bias on the euro remained negative for now, and a correction from a high of $1.3169 hit in mid-September could extend to the $1.24 level, analysts said. Concerns about Greece trumped trade data in China suggesting it is recovering from slower growth, assuaging concerns about the world's second-largest economy. Looking ahead, data due later this week is forecast to show a slowdown in German growth in coming quarters and France slipping into recession. U.S. FISCAL WORRIES Concerns about the U.S. fiscal cliff has driven safe-haven flows into the dollar and kept it near a two-month high against a basket of currencies. It was last little changed at 81.051. Demand for higher-risk currencies has been sluggish as investors fret about the possible impact of some $600 billion in expiring U.S. tax cuts and spending reductions due to begin to take effect in January without a deal on Capitol Hill. Markets on Monday also shrugged off a 0.9 percent July-September quarter-on-quarter contraction in Japan's economic output, which was in line with forecasts. It was the first negative reading in three quarters, and it added to fears that slowing global growth is pushing the Japanese economy into recession. Morgan Stanley held to its medium-term view that the yen would weaken and said it expected dollar/yen to resume its uptrend. The dollar was flat against the yen at 79.45 yen, but above Friday's close of 79.06 yen, its weakest since Oct. 18. The euro was also unchanged against the yen at 100.98.