At 12:16 a.m. (0646 GMT), the partially convertible rupee was at 49.35/36 per dollar, compared with Friday's close of 49.10/11. The market was closed on Monday and will be shut on Thursday on account of religious holidays.
"Investors pulled out funds from the domestic share market on worries the euro zone problem is worsening and on some import buying," said Ashtosh Raina, head of foreign exchange trading at HDFC Bank.
The 30-share BSE index was down 0.4 percent, weighed by high domestic inflation and political uncertainty, while the euro struggled to gain traction against the dollar as debt fears in the euro zone turned to Italy from Greece.
The euro was at $1.3742, compared with $1.3773 in late New York trade on Monday , while the index of the dollar against six major currencies was trading up 0.18 percent at 77.097 points.
Brent crude edged lower as concerns over Italy's sovereign debt overshadowed a potential supply disruption from Iran and seasonally strong demand for fuel.
Oil is India's largest import item and oil refiners are the biggest buyers of dollars in the local market.
"The rupee could fall more in coming days because of a worsening current account deficit and fears of fiscal slippage," Raina said.
Sluggish tax receipts and high refunds have added to the government's fund mismatches, prompting the issue of short-end bills.
India's fiscal deficit reached almost 71 percent of its full-year target in the first half of the year, while the current account deficit widened to $14.1 billion in the June quarter.
The one-month onshore forward premium was at 24.75 points from 26.25 on Friday , the three-month was at 65.75 points from 70.50 at previous close and the one-year was at 182.25 points from 193.
One-month offshore non-deliverable forward contracts were quoted at 49.31, weaker than the onshore spot rate.
In the currency futures market , the most traded near-month dollar-rupee contracts on the National Stock Exchange, the United Stock Exchange and the MCX-SX were all at 49.7. The total volume was at $995.9 million.