Mr Bhandari, a three-term former chief minister, was 77. He is survived by wife Dil Kumari Bhandari, a former MP, two daughters and a son.
He breathed his last at a Delhi hospital this afternoon.
Prime Minister Narendra Modi condoled the demise of Mr Bhandari and praised his role in Sikkim's progress.
"Saddened by the demise of Shri Nar Bahadur Bhandari. His contribution towards Sikkim's progress will always be remembered. My condolences," the prime minister's office said in a twitter post.
Born in 1940 at Malbasey in West Sikkim, Mr Bhandari was a teacher in a government school before he joined active politics.
Known as a pragmati and resilient leader, Mr Bhandari was the second chief minister after L D Kazi to hold the reins of the fledgling state which officially became a part of the Indian Union in 1975.
He was also a former president of Sikkim Pradesh Congress Committee.
Chief Minister Pawan Chamling condoled the death of Mr Bhandari.
"I am extremely sad and shocked to know about the untimely demise of Shri Nar Bahadur Bhandari. With his passing away, Sikkim has lost a worthy son of the soil whose contribution to the state and its people are unparalleld."