Mumbai

You can definitely get admission on LLB (3 years) without having to appear for the entrance exam. There are several law colleges in India that offer admission for LLB on the merit rather than having to appear for the entrance exam. You should look into these universities. I will mention some at the end of the answer.
LLB 3 is becoming rare day by day since most colleges are shifting for the BA LLB course which is 5 years long. There is no difference in the degrees. Both the degrees hold same value when applying for the job.
The difference is that BA LLB is taken after completing class 12th. Candidates have to appear for Common Law Admission Test (CLAT) exam. While for the LLB 3-year, you first need to complete a graduation course. By opting for the first option, you can save 1-year. This is the reason why more and more candidates are preferring BA LLB course.
There are two ways to get admission for LLB course. Firstly, you appear for the entrance exam and secondly is purely based on the merit.
Eligibility
1. A graduation degree in any stream from a recognized college of India.
2. At least 45 percent marks in graduation course.
3. 40 percent marks for SC/ST.
You can apply for colleges below for LLB without having to appear for entrance exam.
1. Government Law College (Mumbai)
In this college, admission is given purely on the merit. The candidate has to go through an interview before admission. There is a cut-off list for admission which varies every year.
2. Balaji Law College (Pune)
For admission, 45 percent marks are required in graduation course. Admission is given on merit.
3. ILS Law College (Pune)
ILS Law College offers both BA LLB and LLB course. For three year course, general a category candidate needs 45 percent while SC/ST need 40 percent marks.