Bardhan, 92, survived by son Ashok and daughter Alka, will be cremated on Monday.
"Bardhan's cremation will take place on January 4 (Monday)," a party statement said.
The CPI leader was admitted to G B Pant hospital here after he suffered a paralytic stroke last month.
Dr Vinod Puri, Director and Professor of Neurology in G B Pant Hospital, said Bardhan "died at 8:20 PM. He suffered a brain stroke due to blockage in the middle cerebral artery. He was in comma."
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"Will always remember Shri AB Bardhan as a passionate Communist, fully committed to his ideology & principles. May his soul rest in peace(sic)," Modi tweeted.
Gandhi said, "In his demise, not only the CPI but the entire nation has lost a leader who championed the cause of THE dispossessed and marginalized throughout his life."
CPI (M) general secretary Sitaram Yechury said Bardhan's wisdom, experience and guidance will be missed.
"Red salute, Comrade Bardhan. We miss your wisdom, experience & guidance (sic)," he tweeted.
JD(U) chief Sharad Yadav said the CPI stalwart had epitomized the continuous struggle for the cause of the masses and was a "tall" leader of the Left movement.
Recalling his long association with Bardhan, Yadav said he first heard him at a public meeting when he was only 12- year-old and went on to build a long political association with him after joining politics.
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.