The Joint Seat Allocation Authority (JoSAA) 2017 is responsible for counselling and seat allocation of candidates who will clear JEE Main 2017 and JEE Advanced 2017.

It is expected that JoSAA 2017 counselling and seat allotment will be conducted in six rounds.

It is expected to be start from last week of June, 2017.

Hi, JEE MAIN counseling shall start in July 2017.
There is no fixed date released.
You can track the dates here:
JEE MAIN 2017 Counselling

All the best.