It has been a rather quiet trading session for the currency market in the absence of any fresh trigger.
The Indian unit traded in a narrow range of 64.02-64.09 most part of the day.
Consistent unwinding by foreign portfolio investors (FPIs) and funds also impacted trading sentiment as they sold shares worth net Rs 383.99 crore yesterday.
The flagship Sensex soared 184 points to close at 33,940.30, while Nifty rose 52.70 points at 10,493.
Meanwhile, global crude prices dropped away from their overnight fresh 2015 highs weighed down by rising US output and the expected January re-opening of the Forties pipeline in the North Sea which triggered traders to lighten up long positions ahead of year-end.
Brent crude, an international benchmark, is trading at $ 64.07 a barrel in early Asian trade.
Globally, the greenback edged higher against other major currencies after US third-quarter data showed the economy grew at its fastest pace in more than two years and also underpinned by the approvals of the most significant US tax code overhaul in three decades.
It later moved in tight range most part of the day before settling at 64.05, showing a mere gain of one paisa.
The RBI, meanwhile, fixed the reference rate for the dollar at 64.0409 and for the euro at 75.8821.
The dollar index, which measures the greenback's value against a basket of six major currencies, was marginally up at 92.93 in early trade.
The local unit, however, drifted back against the Japanese yen to close at 56.50 per 100 yens from 56.43 earlier.
Elsewhere, the British pound traded little changed against the US dollar after the third and final reading for the third quarter GDP headlines the UK economic accelerated to 1.7 per cent in Q3, beating the consensus forecasts of 1.5 per cent.
The euro staged a smart rebound against the US Dollar after the currency experienced a short-term selloff on news of Catalan pro-independence parties winning the absolute majority in regional Parliament in Northern Spain.
In forward market today, premium for dollar remained under pressure due to sustained receiving from exporters.
The benchmark six-month premium payable in May dropped to 117-119 paise from 121-123 paise and the far forward October 2018 contract also moved down to 252-254 paise from 255-257 paise previously.