The Indian rupee is likely to open steady on Tuesday and may look to make mild gains through the session as the dollar's post-U.S. election surge shows signs of cooling after touching a one-year high last week.

The 1-month non-deliverable forward indicated that the rupee will open at 84.38-84.39 to the U.S. dollar, little changed from its close of 84.3850 in the previous session.

The local currency had declined to its record low of 84.4125 last week.

The dollar index was at 106.2 after declining 0.4% on Monday, its second consecutive daily decline, likely on the back of profit-booking after it rose to a one-year peak of 107.07 last week.

"Expect (USD/INR) to move lower, tracking broad dollar weakness but not by much," a foreign exchange trader at a private bank said.

The trader expects the dollar-rupee pair to find immediate support at 84.35-84.36 on the back of hedging demand from importers.

Asian currencies were mostly stronger on the day, led by the Indonesian rupiah, which was up 0.3%.

Meanwhile, the Indian government is likely to extend the term of Reserve Bank of India (RBI) governor Shaktikanta Das for a second time, Reuters reported earlier.

An extension for Das should mean that the "RBI's aggressive FX intervention should still continue for the time being, implying USD/INR (volatility) should still be capped," MUFG Bank said in a note.

Routine interventions by the Indian central bank have supported the rupee for over a month as the currency came under pressure due to sustained portfolio outflows and a post-U.S. election rise in the dollar index and US bond yields.

Overseas investors have pulled out USD 3.3 billion from Indian stocks and bonds on a net basis over November so far, adding to the USD 11.4 billion outflow in the previous month.