The rupee closed at 83.41 against the U.S. dollar, up 0.2% from its close at 83.5550 on Friday.
Indian financial markets were shut on Monday for a holiday.
The rupee rose after drifting in a 10 paisa range last week, with multiple interventions by the Reserve Bank of India that prevented it from falling to its lifetime low of 83.5750, hit in April.
While state-run banks offered dollars early in the session, dollar sales by foreign banks helped the rupee extend its gains, traders said.
At least two large foreign banks sold dollars during the session, likely on behalf of custodial clients, a foreign exchange trader at a large private bank said.
Meanwhile, India's blue chips, the Nifty 50 and BSE Sensex, touched record highs on Tuesday and both ended the day up 0.4%.
The rupee may strengthen beyond 83.20 once inflows from India's inclusion in the JPMorgan emerging market debt index start later this month, said Sajal Gupta, head of forex and commodities at Nuvama Wealth Management's institutional desk.
While foreign investors have been net sellers of Indian equities in June and the year so far, they have bought USD 684 million worth of local bonds this month, taking the tally to USD 7.1 billion of net inflows.
Investors now await U.S. retail sales data later in the day and remarks from a slew of Federal Reserve policymakers.
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