Forex traders said, the domestic unit is trading in a narrow range ahead of the RBI monetary policy decision on Thursday.
At the interbank foreign exchange, the rupee opened at 71.66, then gained some momentum and touched a high of 71.58 against the dollar, showing a rise of 8 paise over its last close.
The Indian rupee on Monday had closed at 71.66 against the US dollar.
Bankers and experts believe the Reserve Bank may cut interest rates for the sixth straight time on December 5 to support growth that has continued to slip.
The RBI has cut interest rates on every single occasion the monetary policy committee (MPC) has met since Shaktikanta Das took over as the Governor in last December.
Traders said rupee is trading in a narrow range as market is awaiting fresh cues on the potential US-China trade deal.
Meanwhile, brent crude futures, the global oil benchmark, rose 0.20 per cent to USD 61.04 per barrel.
Foreign institutional investors remained net sellers in the capital markets, as they sold shares worth Rs 1,731.33 crore on Monday, as per provisional data.
Domestic bourses opened on a cautious note on Tuesday with benchmark indices Sensex trading 66.01 points down at 40,736.16 and Nifty down 23.75 points to 12,024.45.
The dollar index, which gauges the greenback's strength against a basket of six currencies, rose 0.05 per cent to 97.90.
The 10-year government bond yield was at 6.48 per cent in morning trade. DRR DRR DRR