Results on Sunday showed the actress Aishwarya Rai Bachchan, a former Miss World, and her daughter Aaradhya, eight, were infected with coronavirus.
Her husband Abhishek and father-in-law Amitabh, both also actors, were taken to hospital on Saturday with the virus.
Both men were said to have mild symptoms.
Abhishek Bachchan tweeted that they would remain in hospital "till the doctors decide otherwise".
Aishwarya Bachchan, 46, is one of Bollywood's most famous faces both in India and abroad, featuring in several Bollywood and Hollywood films.
She won the Miss World pageant in 1994 and is Goodwill Ambassador for UNAIDS. In 2003 she was the first Indian actress to be a jury member at the Cannes Film Festival.
Aishwarya and her daughter are said to be asymptomatic. Her husband tweeted to say they would be self-isolating at home.
Outpouring of support
On Saturday Amitabh Bachchan told his millions of Twitter followers he had tested positive for Covid-19.
"I have tested Covid positive, shifted to hospital, hospital informing authorities, family and staff undergone tests, results awaited," he wrote.
Bachchan, 77, has been involved in 200 films over five decades.
Bachchan has won multiple awards since rising to prominence in the 1970s. Reuters
He and Abhishek, 44, were taken to Nanavati Hospital in Mumbai on Saturday. Abhishek described them both as having mild symptoms.
Amitabh is currently in the isolation unit of the hospital, news agency ANI reported, quoting a public relations officer for the hospital. He urged anyone who had been close to him in the past 10 days to get tested.
Mumbai municipal officials have since put up banners outside the actor's house in the city, classifying it as a "containment zone".
The news has led to an outpouring of support for the family on social media. Among those paying their respects were actress Sonam K Ahuja and former India cricket player Irfan Pathan.
Well-wishers have been praying for Amitabh Bachchan
"Dear Amitabh ji, I join the whole Nation in wishing you a quick recovery! After all, you are the idol of millions in this country, an iconic superstar! We will all take good care of you. Best wishes for a speedy recovery!" said India's Health Minister Harsh Vardhan.
Bachchan Snr has enjoyed starring roles in hit movies such as Zanjeer and Sholay. Since rising to fame in the 1970s, he has won numerous accolades including four National Film Awards and 15 Filmfare Awards. France has also bestowed its highest civilian award, the Legion of Honour, for his contribution to cinema.
Outside acting, Bachchan Snr had a brief stint in politics and was elected as a member of India's parliament in 1984 at the behest of former Prime Minister Rajiv Gandhi. But he resigned three years later, disillusioned by a corruption scandal under Mr Gandhi's government.
In recent months, he has been prominent in helping the government get its message across in the fight against coronavirus.
India saw a record rise in the number of coronavirus cases by 27,100 on Sunday, with the total climbing to nearly 850,000 - the third highest caseload in the world. There have been complaints about a lack of both testing and frontline medical staff.
Prime Minister's tweet
Meanwhile, Prime Minister Mahinda Rajapaksa in a Twitter message on Sunday wished the Bachchan family a speedy recovery.
Speaking at a public meeting held at Lunugamvehera in Tissamaharama yesterday, former UNP Minister Harin Fernando said that Mahinda Rajapaksa was more concerned about the well being of Aishwarya Rai than the many hardships faced by the people of Sri Lanka during this global pandemic.
"Around 80 families have been quarantined in Rajanganaya. They are faced with many serious issues. These families can't make ends meet as they are unable to make a daily income when they are being quarantined. The entire family ends up destitute. None of the government institutions are paying attention to them. This scenario applies to many families that have been quarantined in rural areas. The 'wanna be' Prime Minister doesn't care at all. He feels sorry for Aishwarya... '' Harin Fernando said.
Fernando also took to social media with the following tweet.