In lacklustre activity on the Interbank Foreign Exchange (Forex) market, the Indian unit opened steady at 39.39/40 per dollar from Wednesday's close of 39.3850/3950 per dollar.
Later, it fell and was quoted at 39.4150/4250 a dollar in late morning deals.
Month-end dollar demand from banks on behalf of their clients, mainly oil refiners, to meet the import requirements weighed on the rupee, dealers said.
Global crude oil prices hovered near USD 92 a barrel in Asian trade on Thursday.
Sluggish equity markets also pushed the rupee downwards.
The Indian benchmark Sensex tumbled by 423 points to 17,417.63 in late morning trade on Thursday, while most of the Asian markets displayed a narrowly mixed trend.
The US Federal Reserve on Wednesday cut the lending rate by 0.50 per cent to prevent the economy from sinking into recession but could not stem the fall in dollar against major rivals.
Foreign Institutional Investors (FIIs) pulled out nearly USD 4 billion from equities since January 16, which also impacted negatively on the rupee sentiment.