Gaining some ground, the rupee recovered against the euro at 70.97 from 71.25 previously but fell against the pound sterling to 100.29 from 100.09 earlier.
The domestic unit opened lower at 63.80 against the US dollar at the Interbank Foreign Exchange (Forex) and dropped sharply to 63.94 on heavy demand for the American currency before ending down by 20 paise, or 0.31 per cent, at 63.84.
It moved in a range of 63.80 and 63.94 per dollar during the day.
The dollar index was trading lower by 0.60 per cent against its major global rivals today.
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In New York market, the US dollar ended higher against the basket currencies, after Greece failed to strike a deal with its lenders, taking it a step closer to a debt default and exit from the Eurozone.
Meanwhile, oil prices in Asia fell today, tracking the sell-off in global equity markets on fears of a Greek debt default, analysts said.
The benchmark BSE Sensex recovered from initial sharp losses but ended lower by 167 points, or 0.60 per cent, on all-round selling pressure from operators as stocks fell across the globe on worsening economic crisis in Greece and odds of a default increasing after the country announced plans for a referendum on its bailout conditions..