The former Superior General of Missionaries of Charity was suffering from various ailments.
She took over the running of the charity after Mother Teresa died in 1997 and ran it for 12 years before Sister Mary Prema took over in 2009.
The Missionaries of Charity cares for the homeless and dying in Kolkata.
Archbishop of Kolkata Thomas D'Souza told the AFP news agency that Sister Nirmala "passed away peacefully, surrounded by sisters praying for her".
"She had been suffering from a heart disease for quite some time, but she never lost her smile and was always cheerful."
He said Sister Nirmala "carried forward Mother Teresa's legacy of compassion, gentleness, service to the poorest of the poor and holiness of life".
"Even after she ceased to be head of Missionaries of Charity, she served the people with the same zeal as she did before."
Reports say Sister Nirmala's body will be kept at the Missionaries of Charity headquarters in Kolkata and her funeral will be held on Wednesday afternoon.
Prime Minister Narendra Modi led the tributes to the deceased nun.
West Bengal Chief Minister Mamata Banerjee said she was "saddened" by Sister Nirmala's death.
The Missionaries of Charity has about 4,500 nuns and hundreds of homes around the world.
Sister Nirmala, the nun who succeeded Mother Teresa as the head of a charity in the Indian city of Kolkata (Calcutta), has died, aged 81.