Thanks for the question. 

Private institutions do take admissions on the jee mains rankings whether or not you qualified. 

You have to register yourself separately on each institution's website  then they prepare their merit list and you are called accordingly.

few colleges are-

MHP

ACE College of Engineering

University of Kerala & KTU

Thiruvallam, Trivandrum

2013

Manarul Huda Trust

2

ASI

Adi Shankara Institute of Engineering Technology

Mahatma Gandhi University & KTU

Kalady, Ernakulam

2001

Adi Shankara Trust

3

ATP

Ahalia School of Engineering and Technology

University of Calicut & KTU

Kozhippara, Palakkad

2012

Ahalia International Foundation

4

AAE

Al Azhar College of Engineering and Technology

Mahatma Gandhi University & KTU

Thodupuzha,Ernakulam

2010

Noorul Islam Trust Al-Azhar Group of Institutions

5

AAP

Al-Ameen Engineering College

University of Calicut & KTU