The rupee edges up to 55.42/43 versus its previous close of 55.535/545, tracking the greenback's losses versus other Asian currencies and a slight risk-on mood in global markets.
Asian shares inch up on Monday on hopes that Greece can avoid a near-term bankruptcy.
Euro at a seven-month high against the yen and a one-month peak against the dollar.
Nifty India stock futures traded in Singapore up 0.3 percent, suggesting a positive start to the local market.
Traders expect oil-related dollar demand to meet month-end import commitments to prevail through the day, limiting any sharp fall in the pair. USD/INR seen moving in a 55.30 to 55.60 range initially.