The rupee falls to 54.3525/3600, tracking negative local shares and heavy losses in the euro in Asian trade.
The pair last closed at 54.175/185.
Traders said a few stopgaps were getting triggered, which prompted dollar buying.
The Sensex was down 0.24 percent, while the Nifty was down 0.16 percent, after the two indices fell for a seventh consecutive session on Monday.
The euro was hit by fears that future bank rescues in the euro zone would come with the same stern conditions seen in Cyprus' deal.