* Euro struggles on report Greece wants to amend EU deal
* Fed minutes suggest rates could stay low for some time
* Aussie rallies after RBA raises rates, flags more hikes
* Canadian dollar hits parity with U.S. unit

(Adds details, updates prices)

By Steven C. Johnson

NEW YORK, April 6 (Reuters) - The euro fell on Tuesday as reports Greece wanted to amend a European Union aid deal rekindled fear about its ability to resolve its debt crisis.

The surging Canadian dollar hit parity with its U.S counterpart for the first time since mid-2008 as rising commodity prices boosted the outlook for higher Canadian interest rates.

U.S. interest rates, on the other hand, could stay low for even longer than investors have been anticipating, minutes from the Federal Reserve's last meeting showed. That helped send the Australian dollar, whose central bank raised interest rates on Tuesday, to its highest against the greenback since January.

The euro fell broadly, at one point dipping below $1.34 after media reported Greece wanted to renegotiate a joint EU-International Monetary Fund aid deal reached last month. [ID:nLDE63513Y]

Greek banks were hit as big depositors moved cash overseas, and the euro fell as investors ditched Greek assets, pushing up the risk premium on Greek government bonds.

Greece denied the reports, but this did little to quell concern. The yield spread between 10-year Greek and German government bonds at one point exceeded 4 percentage points, the widest since the euro's launch.GR10YT=TWEBEU10YT=TWEB

There is no doubt that the Greek financing story continues to plague the euro, said Boris Schlossberg, director of research at GFT Forex in New York.

If Greece has trouble selling debt in the coming weeks, he said the euro could fall further on renewed concerns over the viability of monetary union.

The euro fell against the dollar for a third straight day, dropping 0.6 percent to $1.3402. It has fallen 6.5 percent so far this year against the dollar. The euro was also down 1.2 percent at 125.66 yen EURJPY=, while the dollar fell 0.6 percent to 93.70 yen.

Greece needs to sell more bonds to meet its funding needs. It has raised about 23 billion euros of a projected 2010 requirement of 53.2 billion euros. Last month, it announced plans to sell a dollar bond at the end of April.

FED ON GUARD, CANADIAN, AUSSIE DOLLARS GAIN

The dollar wobbled a bit after minutes showed the Fed was still nursing some concerns about the U.S. economy and in no hurry to raise rates from record lows. [ID:nWEQ003852]

Though it only pared gains against the euro, the dollar lost ground against currencies where interest rates have either already risen or are expected to rise soon.

The Canadian dollar rose to C$0.9988 per U.S. dollar, breaking parity for the first time since July 2008, before easing to C$1.0012 per greenback, little changed from Monday.

Fergal Smith, market strategist for Canada at Action Economics in Toronto, said the loonie could rise as high as C$0.98 per U.S. dollar as Canada's economic outlook improves and markets gear up for tighter monetary policy.

The pressure will remain on (the Bank of Canada) to turn more hawkish at its next policy announcement, he said.

The Reserve Bank of Australia is already well into its own tightening policy, and its move to hike the cash rate to 4.25 percent pushed the Australian dollar to $0.9288, its highest since January. [ID:nSGE6340JG]

The market was also watching the yuan after U.S. Treasury Secretary Timothy Geithner said he was confident China would see that it is in its own interest to make its currency more flexible. [ID:nSGE6350GP]

Dollar/yuan six-month offshore forwards CNY6MNDFOR= fell to their lowest since August 2008 on Tuesday, implying greater future yuan appreciation. [ID:nTOE63506L]

(Additional reporting by Wanfeng Zhou; Editing by Andrew Hay)