The rupee is still at 61.15/16 vs previous close of 61.34/35, though off lows of 60.9850.
Local shares are also off highs, trading with 0.2% gain.
"The fall in stocks and the euro giving up gains has led to the rupee's fall. Moreover, the month-end demand from state-run banks for their clients seems to have started," said a dealer with a private bank.
Foreign investors bought Indian shares worth $117.87 million on Thursday when the benchmark BSE index fell 0.4%, provisional exchange data shows, highlighting the strong overseas interest for Indian equities.
Most emerging Asian currencies were down on Friday, hit by uncertainty about the outlook for Federal Reserve policy and as a beleaguered Chinese yuan braced for its worst weekly loss in more than two decades.