"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.