As you have asked about the coaching institutes for IIT JEE exam in Chennai, I am mentioning some of them, check below for the details

IIT-Jee Coaching Centre
Special Education School
Address: Plot No. 2797, Old Y Block, 12th Main Road, 6th street, Anna Nagar, Chennai, Tamil Nadu 600040
Phone:044 2621 5550

PRAGNA CLASSES LLP
Coaching Center
Address: Vedanta Desigar Swami St, Vinayaka Nagar, Mylapore, Chennai, Tamil Nadu 600004
Phone:095000 14341

Fiitjee Towers
Coaching Center
Address: No 3, Fiitjee Towers, First Lane, Nungambakkam High Road, Chennai, Tamil Nadu 600034
Phone:044 4393 7100

Anugraham Classes
Tutoring Service
Address: Manikkam St, Ramakrishnapuram, West Mambalam, Chennai, Tamil Nadu 600033
Phone:044 2474 9441

Emphasis IIT JEE Academy
Coaching Center
Address: # 37/112 Gollavar Agraharam Road, Sanjeevaraya Pooram, Old Washermenpet, Chennai, Tamil Nadu 600021
Phone:086809 92557

TRS IIT CLASSES
Coaching Center
Address: 37/72, 7th Avenue, Ashok nagar, (opp. GRT School Play Ground),, Chennai, Tamil Nadu 600083
Phone:094443 61199

Aakash Institute
Coaching Center
Address: No. 2, AB-Block, 2nd Avenue, Anna Nagar, Chennai, Tamil Nadu 600040
Phone:044 3945 4545

FIITJEE
Coaching Center
Address: 175, Poonamallee High Road, Opp Ega Theatre, Kilpauk, Chennai, Tamil Nadu 600010
Phone:044 4285 9701

Career Launcher
Coaching Center
Address: 108, 2nd Floor, Rajalakshmi Mahal, Near Panagal Park, Sir Thyagaraya Rd, T Nagar, Chennai, Tamil Nadu 600017
Phone:044 2815 4725