Traders said month-end demand for dollars from importers, particularly oil firms, is expected to keep a downward bias on the rupee over the week but foreign fund flows will be crucial for direction.
Most emerging Asian currencies rose on Monday after a stronger-than-expected Chinese manufacturing survey bolstered hopes of growth in the world's second-largest economy.
Traders will continue to monitor global crude oil prices and stocks for clues.
"Dollar demand from importers hurt the rupee today. But the market will continue to be rangy until the budget. The rupee will remain in a broad 59.50 to 60.50 range until then," said Ashtosh Raina, head of foreign exchange trading at HDFC Bank.
Indian finance minister Arun Jaitley will present the federal budget for 2014/15 on July 10, government sources said.
The partially convertible rupee closed at 60.20/21 per dollar compared to 60.1850/1950 on Friday. The rupee moved in a range of 60.05 to 60.2750 in the session.
Indian shares fell for a fourth straight session on Monday, marking their lowest close in 2-1/2 weeks as ITC Ltd slumped the most in ten months on tax worries, while concerns over high oil prices feeding inflation continued to weigh.
In the offshore non-deliverable forwards, the one-month contract was at 60.55 while the three-month was at 61.14.