India is home to millions of Tamils in the southern state of Tamil Nadu, which is separated by a narrow strip of sea from Sri Lanka.
About 74,000 Sri Lankan Tamils are also housed in refugee camps in the state.
"As conventional conflict in Sri Lanka comes to a close, this is the moment when the root causes of the conflict in Sri Lanka can be addressed," foreign ministry spokesman Vishnu Prakash told reporters in New Delhi.
He said India wanted "effective devolution of power within the Sri Lankan constitution so that Sri Lankans of all communities including the Tamils can feel at home and lead lives of dignity of their own free will.
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Resentment of the minority Tamils' privileged status in Sri Lanka under British colonial rulers surfaced when the Sinhalese majority took power after the island gained independence in 1948.
With Sinhalese nationalism on the rise, Tamil youngsters took up arms and a full scale war erupted in the 1980s.
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Prakash said that India expected Sri Lanka to