Forex traders said the rupee is trading in a narrow range as market participants are awaiting cues from the Federal Open Market Committee (FOMC) meeting on July 31.
Moreover, strengthening of the US dollar vis-a-vis other currencies overseas kept pressure on the Indian rupee, they added.
The rupee opened weak at 68.73 at the interbank forex market and then fell further to 68.79, down 4 paise over its last close.
The rupee had settled at 68.75 against the US dollar on Monday.
Besides, market participants are also looking forward to the US and China trade talks, which can impact rupee movement.
The US and Chinese officials are restarting negotiations in Shanghai on Tuesday in an effort to resolve the year-long trade dispute.
Meanwhile, a positive opening in domestic equities supported the local unit and restricted the fall.
Brent crude futures, the global oil benchmark, rose 0.49 per cent to USD 64.02 per barrel.
Foreign institutional investors (FIIs) remained net sellers in the capital markets, pulling out Rs 704.42 crore on Monday, as per provisional data.
Domestic bourses opened on a positive note day with benchmark indices Sensex trading 252.30 points up at 37,942.91 and Nifty up 74.35 points at 11,263.55.