These are both very good options for Mechanical Engineering, however, there are a few things to consider when deciding which one is best for you.
Location: IIT Delhi is about 70 kms from the center of the capital, New Delhi, which is known for its excellent transportation, lifestyle, job opportunities and reasonable cost of living. IIT Bombay, on the other hand, is located in Mumbai, one of the biggest cities in India. Both cities also offer a number of cultural, educational and leisure activities.
Cost of Education: IIT Delhi is one of the cheapest Indian Institutes of Technology (IITs). Its four year B.Tech program costs approximately Rs. 5-6 lakh, while IIT Bombay's four year B.Tech program costs around Rs. 8.5-9.5 lakh.
Faculty: IIT Delhi has a highly experienced faculty with many of them having worked at IITs before. The faculty at IIT Bombay is also renowned, with many world-renowned professors.
Curriculum: Both IIT Delhi and IIT Bombay offer a diverse curriculum in Mechanical Engineering. The courses are designed to develop core technical knowledge and to enable a wide range of career options for students.
Research: IIT Delhi is known for its outstanding research capabilities. It has produced many renowned scholars in the field. However, IIT Bombay is known for its industry collaborations, which offer students a variety of career opportunities.
Ultimately, the best institute for you will be the one which best meets your personal interests, financial capabilities and career goals. You should consider these points and make an informed decision.