IIT Bhubaneswar offers VLSI as one of the electives in the course curriculum of M.Tech in Electronics and Communication Engineering. The program is quite good both in terms of academics and placement opportunities. According to NIRF, IIT Bhubaneswar is ranked 22 among top engineering institutes. The admission is done through GATE score and the total course fee is 4.62 lakhs. In 2019, a total of 69% of students from the M.Tech courses got job offers and the average CTC was around 11.4 LPA. The top recruiters were HPCL, IOCL, Oil India Limited, Goldman Sachs, GE India, Honeywell, Infosys, TCS, R&D, Adobe.
Anna University offers M.E in VLSI. According to NIRF Ranking 2020, it is ranked 20 among the top universities of India. The selection is based on the TANCET or GATE score followed by counseling (TANCA). The total course fee is 1.2 lakhs. Some of the top recruiters include Caterpillar, L&T, Samsung Semiconductor R&D, Wipro Limited (VLSI and Telecom), Qualcomm, Renault Nissan, HCL, Dell, Sony, Bharat Electronics Limited, LG, etc. The median salary offered to the students of this program is around 4.2 LPA.
It is on you to determine which university will be best for you.