Singh, a former Army officer, had been ill after a fall at his home in August 2014 and was admitted to the Army Research and Referral Hospital. He had been in and out of the hospital and was admitted again in June this year.
"It is with profound grief that we inform about the sad demise of Hon'ble Major
Despite the best efforts of the attending team of specialists to resuscitate him, he could not be revived and passed away, it added.
His funeral will take place later today at Jodhpur in Rajasthan, family sources said.
Condoling his death, "Modi said, "Jaswant Singh Ji served our nation diligently, first as a soldier and later during his long association with politics. During Atal Ji's Government, he handled crucial portfolios and left a strong mark in the worlds of finance, defence and external affairs. Saddened by his demise."
Singh, considered close to Vajpayee and veteran leader
He had fought the 2014 Lok Sabha election as an independent candidate after the BJP did not give him ticket, but lost.
In his message, Modi added, "Jaswant Singh Ji will be remembered for his unique perspective on matters of politics and society. He also contributed to the strengthening of the BJP. I will always remember our interactions. Condolences to his family and supporters. Om Shanti."
Expressing his anguish, Defence Minister Rajnath Singh said Jaswant Singh served the nation in several capacities and distinguished himself as an effective minister and parliamentarian.
"Jaswant Singh ji would be remembered for his intellectual capabilities and stellar record in service to the nation. He also played a key role in strengthening the BJP in Rajasthan. Condolences to his family and supporters in this sad hour. Om Shanti," Singh said.