Hello Candidate,
You need to write centralised entrance exams to get a seat at NITs, IITs and GFITs. However private universities and colleges offering BTech also have their own entrance exams and management quota.
Almost all private universities and colleges have 5-10% seats reserved for management quota.
Some of the best private Institutes of Mumbai are:
MIT world peace University, pune.
MIT.
KJ Somaiya college of engineering.
Vivekananda institute of technology.
You can read more here : https://engineering.careers360.com/colleges/list-of-private-engineering-colleges-in-mumbai