Some of the best Technical Institutes/Research Organizations for Integrated M.Tech+PhD in Chennai are as follows

1. IIT Madras

https://www.careers360.com/university/indian-institute-of-technology-madras/amp

2. AcSIR Chennai

https://www.careers360.com/university/academy-of-scientific-and-innovative-research-chennai/mtech-and-phd-engineering-integrated-course

3. CMI Chennai/ Institute of Mathematical Sciences, Taramani, Chennai (Mathematics/Statistics/Computational Studies)

For most of the Institutes, as a Basic Requirement you need GATE Scorecard, followed by respective Centralised admission procedures by Respective Institute as far as M.Tech+PhD is concerned. As far is AcSIR is concerned, you even require the CSIR UGC NET Score for Admission Purposes.