"I have tested Covid positive, shifted to hospital, hospital informing authorities, family and staff undergone tests, results awaited," he wrote.
His son, Abhishek, later tweeted he had also tested positive.
Bachchan, 77, has been involved in 200 films since becoming a star five decades ago.
He, and Abhishek, 44, have been taken to Nanavati Hospital in Mumbai.
Amitabh urged anyone who had been close to him in the past 10 days to get tested.
A hospital official told The Indian Express newspaper that he was in a stable condition.
His son described them both as having mild symptoms.
Bachchan Snr has enjoyed starring roles in hit movies such as Zanjeer and Sholay.
His latest film was the comedy Gulabo Sitabo, released on Amazon.
He has also been prominent in helping the government get its message across in the fight against coronavirus.
India has seen a sharp rise in the number of coronavirus cases, with the total climbing to nearly 821,000 on Saturday - the third highest caseload in the world.
There have been complaints about lack of testing and frontline medical staff.