The rupee pared initial gains to settle 2 paise lower at 75.58 against the US dollar on Friday amid firming oil prices and lacklustre global cues.
Forex traders said the local currency was trading in a narrow range amid rising risk-aversion in the broader financial markets.
Investors also awaited further details of the government's fiscal stimulus package, they added.
At the interbank foreign exchange market, the rupee opened at 75.51, and stayed in a range of 75.45 - 75.59. The local unit finally finished at 75.58, down 2 paise over its previous close.
On Thursday, the rupee had settled 10 paise lower at 75.56 against the greenback.
On a weekly basis, the domestic unit has depreciated by 4 paise. It had settled at 75.54 on May 8.
Announcing the third tranche of stimulus measures for the coronavirus-hit economy, Finance Minister Nirmala Sitharaman said this will deal with giving relief to agriculture and allied industries.
Forex traders said market participants are concerned about the implications of the Rs 20 lakh crore economic stimulus package on the fiscal deficit, as there is still no clarity on how it would be financed.
Meanwhile, domestic bourses ended modestly lower after a choppy session. The 30-share BSE Sensex settled 25.16 points or 0.08 per cent lower at 31,097.73, while the NSE Nifty slipped 5.90 points, or 0.06 per cent, to 9,136.85.
Foreign institutional investors remained net sellers in the capital market, selling equity shares worth Rs 2,152.52 crore on Thursday, according to provisional exchange data.
"Overall market sentiments have been lacklustre and this week, the spot market traded in a very tight range. The fiscal measures undertaken by FM Sitharaman to reduce risk aversion and improve liquidity didn''t have major impact on USD/INR spot," said Rahul Gupta, Head of Research- Currency, Emkay Global Financial Services.
Brent crude futures rose 1 per cent to $31.44 per barrel.
"Gold and crude both scaling up give weakness to the rupee prices as India being net-importer of both. Rupee scaled low to 75.60 taking resistance near 75 against the US dollar," said Jateen Trivedi, Senior Research Analyst (Commodity & Currency) at LKP Securities.
Traders further said investors are also concerned about rising coronavirus cases and the impact on the domestic as well as global economy.
The US dollar strengthened overseas as fears of a second wave of coronavirus infections prompted investors to dump risky assets.
In India, the death toll due to COVID-19 rose to 2,649 and the number of cases climbed to 81,970, according to the health ministry.
The number of cases around the world linked to the disease has crossed over 44.44 lakh and the death toll has topped 3.02 lakh.
The Financial Benchmark India Private Ltd (FBIL) set the reference rate for the rupee/dollar at 75.4440 and for rupee/euro at 75.4440. The reference rate for rupee/British pound was fixed at 75.4440 and for rupee/100 Japanese yen at 70.64.