* Euro falls on fresh worries about Greece and Spain * Single currency hits 3-week high versus Swiss franc * Dollar index up 0.3 percent By Jan Harvey LONDON, Oct 9 (Reuters) - The euro fell on Tuesday, pulling further away from recent highs as uncertainty about when Spain will apply for a bailout and fresh concerns about Greece weighed on investor sentiment. European Central Bank President Mario Draghi's reiteration that the ECB's financial support for troubled euro zone states was unlimited within certain conditions helped limit losses. The euro was 0.3 percent lower on the day at $1.2932, well below a two-week high of $1.3072 hit on Friday but off a session low of $1.2907. Traders cited bids below $1.2900 and offers around $1.3000, suggesting a narrow range for the euro. Against the yen, it was down 0.3 percent to 101.24 yen , well below Friday's two-week high of 102.80 yen, as some Japanese investors exited their long euro/yen positions. ING Investment Management fund manager Jaco Rouw said he remained "cautious" on the euro, attributing the bulk of its gains since late July to dollar weakness on the back of U.S. quantitative easing measures announced by the Federal Reserve. "Going forward, we think the euro zone economy is clearly one of the weakest in the world, and we think the ECB will have to cut rates further," he said. "Although important steps have been made both by the ECB and European politicians to save the euro, we think we will get new periods of stress. Markets are currently in a bit more of a complacent stage, so also there we think risks are to the downside," he added. Draghi told the European Parliament Committee on Tuesday that Greece has made progress on reforming its economy, but has more work to do. Overall, the euro zone economy faced a long and uphill road to recovery and the bloc was still facing a crisis of confidence, he added. Those comments came after euro zone finance ministers and the International Monetary Fund held a "thorough and robust" debate on Greece, but failed to make significant progress in deciding how best to get the country back on track with its bailout programme. Late in September, Reuters reported Greece's international lenders were at loggerheads over how to solve Athens' debt crisis with the IMF demanding European governments write off some of the Greek debt they hold. Greek Finance Minster Yannis Stournaras said international lenders were considering its request to give Greece two more years to reach its budget deficit reduction targets, but added they had just started discussions. "While uncertainty about Spain plays out, investors are also getting worried about the Greek situation," said Adam Myers, senior currency strategist at Credit Agricole. Euro zone finance ministers defended Spain on Monday, saying the country was taking steps to overhaul its economy, successfully funding itself in the financial markets, and did not need a bailout, at least for now. SWISS FRANC DROPS Earlier the euro hit a three-week high against the Swiss franc as traders cited a media report that a U.S. custodial bank was set to impose a penalty on Swiss and Danish crown deposits. Both currencies have been safe havens for investors fleeing the euro zone debt crisis. While the Swiss National Bank imposed a floor on the euro/Swiss franc pair at 1.20 francs to curb inflows, Denmark's central bank has cut official rates. "The Swiss franc weakened after Bloomberg reported that a custodian bank plans to impose negative interest rates on Swiss and Danish crown deposits from Nov. 1," Gareth Berry, currency strategist at UBS Singapore, wrote in a note. He said the franc's weakness was likely to be temporary given rates have been negative in the interbank market for months, and that those wanting to hold Swiss franc deposits for safe haven reasons were unlikely to be deterred by the penalty. The euro was 0.1 percent higher at 1.2109 francs, having risen to a high of 1.21435 earlier in the day. The dollar was also 0.4 percent higher against the Swiss franc, trading at 0.9366 francs. The dollar's index against a basket of currencies was up 0.3 percent at 79.746, well above Friday's two-week low of 79.103. Still it needed to rise convincingly above the Oct. 1 high of 80.147 to start a fresh uptrend. Against the yen, the dollar was little changed at 78.28 yen , but still below a two-week high of 78.88 set on Friday.

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