Mr Bardhan, 92, is survived by son Ashok and daughter Alka. His wife, a professor in Nagpur University, died in 1986. Mr Bardhan was admitted to Delhi's GB Pant hospital after he suffered a paralytic stroke last month.
Dr Vinod Puri, Director and Professor of Neurology in G B Pant Hospital, said Mr Bardhan "died at 8:20PM. He had a brain stroke due to blockage in the middle cerebral artery. He was in comma".
Earlier in the day, Mr Raja said the condition of Mr Bardhan had "worsened" with his blood pressure levels falling.
"Condition of Comrade Bardhan has worsened today. He was put off ventilator yesterday and was able to breath normally. But today, his blood pressure (level) fell and his condition has turned very critical now," CPI national secretary D Raja said.
Mr Bardhan, who lived in the CPI headquarters in the national capital, was admitted to the hospital on December 7 after he felt uneasiness and lost consciousness.
Mr Bardhan has been a leading figure of the trade union movement and Left politics in Maharashtra and a key figure in fostering relations between the Congress-led UPA and the Left in the early 2000s.
Mr Bardhan entered the electoral politics in 1957 when he won as an Independent candidate in Maharashtra Assembly polls but there failed to achieve success in Lok Sabha and Rajya Sabha polls.
He later rose to become the General Secretary and then President of All India Trade Union Congress, the oldest trade union in India.
Among those who paid tributes to Mr Bardhan were Prime Minister Narendra Modi, Communist Party of India-Marxist general secretary Sitaram Yechury, BJP leader Mukhtar Abbas Naqvi and West Bengal Chief Minister Mamata Banerjee.
Will always remember Shri AB Bardhan as a passionate Communist, fully committed to his ideology & principles. May his soul rest in peace.- Narendra Modi (@narendramodi) January 2, 2016
Red salute, Comrade Bardhan. We miss your wisdom, experience & guidance, Comrade. Pledge to carry forward the Indian revolution.- Sitaram Yechury (@SitaramYechury) January 2, 2016
Demise of CPI leader A B Bardhan huge loss to Indian politics,My deep condolences to his family & well wishers.- Mukhtar Abbas Naqvi (@naqvimukhtar) January 2, 2016
Saddened at the passing of veteran politician AB Bardhan. Condolences to his family and colleagues in his party- Mamata Banerjee (@MamataOfficial) January 2, 2016