As you have asked about the colleges in Chennai offering admission through Tamil Nadu Engineering Admission (TNEA), I am mentioning them, check below for the details

Aalim Muhammed Salegh College of Engineering, Chennai
Aarupadai Veedu Institute of Technology, Kanchipuram
Agni College of Technology, Chennai
B.S. Abdur Rahman University (B.S. Abdur Rahman Crescent Engineering College), Chennai
Bharath University (Bharath Institute of Higher Education and Research), Chennai
Chennai Institute of Technology, Chennai
Dhanalakshmi College of Engineering, Chennai
Dr. MGR University (Dr. M.G.R. Educational and Research Insitute), Chennai
Easwari Engineering College, Chennai
G.K.M. College of Engineering and Technology, Chennai
Hindustan University (Hindustan College of Engineering), Chennai
Jerusalem College of Engineering, Chennai
K.C.G. College of Technology, Chennai
Meenakshi Sundararajan Engineering College, Chennai
PERI Institute of Technology, Chennai
Raja Rajeswari Engineering College, Chennai
S.A. Engineering College, Thiruverkadu
S.K.R. Engineering College, Thiruvallur
Sree Sastha Institute of Engineering and Technology, Chennai
Sri Lakshmi Ammal Engineering College, Chennai
Sri Sai Ram Institute of Technology, Chennai
Sri Sairam Engineering College, Chennai
St. Peter's University (St. Peters Engineering College), Chennai
T.J. Institute of Technology, Chennai
Tagore Engineering College, Chennai
Thangavelu Engineering College, Chennai
Central Institute of Plastic Engineering and Technology (CIPET), Chennai
University Departments of Anna University Chennai, ACT Campus, Chennai
University Departments of Anna University Chennai, CEG Campus, Chennai
University Departments of Anna University Chennai, MIT Campus, Chennai
Alpha College of Engineering, Chennai
Anand Institute of Higher Technology, Chengalpattu Taluk
Annai Veilankanni's College of Engineering, Chennai
Central Institute of Plastics Engineering and Technology, Chennai
Davinci School of Design and Architecture, Chennai
Dhaanish Ahmed College of Engineering, Chennai
Gojan School of Business and Technology, Chennai
Gopal Ramalingam Memorial Engineering College, Chennai
Jawahar Engineering College, Chennai
Jeppiaar Engineering College, Chennai
Loyola-ICAM College of Engineering and Technology, Chennai
Madha Engineering College, Chennai
Madha Institute of Engineering and Technology, Chennai
Magna College of Engineering, Tiruvallur
Measi Academy of Architecture, Chennai
Meenakshi College of Engineering, Chennai
Misrimal Navajee Munoth Jain Engineering College, Chennai
Mohamed Sathak A.J. College of Engineering, Chengelpet Taluk
New Prince Shri Bhavani College of Engineering and Technology, Chennai
P.M.R. Engineering College, Chennai
Panimalar Engineering College, Chennai
Panimalar Institute of Technology, Chennai
Prince Dr. K. Vasudevan College of Engineering and Technology, Chennai
Prince Shri Venkateshwara Padmavathy Engineering College, Chengalpet
S.R.R. Engineering College, Chengalpattu
Sathyabama University (Sathyabama Engineering College), Chennai
Sree Sastha College of Engineering, Chennai
Sri Krishna Engineering College, Sriperumpudur
Sri Krishna Institute of Technology, Chennai
Sri Muthukumaran Institute of Technology, Chennai
Sri Ramanujar Engineering College, Chennai
Sri Sivasubramaniya Nadar College of Engineering, Chennai
Srinivasa Institute of Engineering and Technology, Chennai
St. Joseph's College of Engineering, Kanchipuram
St. Peter's College of Engineering and Technology, Chennai
University Departments of Anna University Chennai, SAP Campus, Chennai
Vel Tech, Chennai
Vel Tech Dr.RR and Dr.SR Technical University (Vel Tech Engineering College), Chennai
Vel Tech High Tech Dr.Rangarajan Dr.Sakunthala Engineering College, Chennai
Vel Tech Multi Tech Dr.Rangarajan Dr.Sakunthala Engineering College, Chennai
Velammal Engineering College, Chennai