The Indian movie icon said in a message posted on Saturday evening that he had been “shifted to hospital” after contracting the disease.
He added: “Family and staff undergone tests, results awaited.
“All that have been in close proximity to me in the last 10 days are requested to please get themselves tested.”
His son Abhishek, 44, also an actor, then tweeted that he, too, had been struck down with the virus.
He said that both he and his father have “mild symptoms”, adding: “I request all to stay calm and not panic.”
The elder Bachchan, who is a household name in India, has previously been named among the highest paid actors in the world on Forbes’ rich list.
In 2015, he was present for the official unveiling of a statue of Mahatma Gandhi in Parliament Square alongside then-Prime Minister David Cameron.
India is currently in the midst of a sharp rise in coronavirus cases.
On Saturday the country reported a new daily high of 27,114 cases. Meanwhile, a partial lockdown has been imposed in several high-risk areas.