The talks are also being held with Malaysia's IHH Healthcare Bhd, India's Manipal Hospitals and Fortis Healthcare, the sources added, declining to be identified because the information has not been made public.
"The formal process is on. We have seen initial interest and this will take a few months to conclude," said one of the sources.
Advent, Carlyle and IHH declined to comment, while Manipal and Fortis did not respond to Reuters' emails seeking comments.
Advent bought its 72 per cent stake in CARE Hospitals in 2012 for $105 million. The chain operates 17 hospitals in nine cities across India. The remaining 28 per cent stake is currently held by a group of Indian doctors.
Interest in Indian private hospital chains has increased as public spending in healthcare has remained low, resulting in a dilapidated network of government hospitals and clinics, especially in rural areas.
IHH has been especially active in looking to invest in India. Last month, it bought a 51 per cent stake in India's Continental Hospitals Ltd for $45.4 million, and also owns an 11 per cent stake in the largest Indian hospital chain, Apollo Hospital Enterprise.
© Thomson Reuters 2015