Chennai has one of the best hospitals in India. It is the hub for medical tourism in South Asia. There are multiple hospitals in Chennai that are thronged by foreigners for medical treatment as the costs in India are extremely low compared to the costs in the West. You've heard accurate facts on that.
In order of popularity I recommend the following hospitals taking into account the fact that you need English speaking staff:
Apollo Hospitals- Greams Road and Anna Salai;
Global Hospitals, OMR;
Kauvery Hospital, Alwarpet;
St. Isabel Hospital, Mylapore;
Chettinad Health City, OMR.
Hope the answer helped you.