Hello Aspirant. You mentioned  your AIR in NEET PG is 40444 and you belong to BC (or OBC) category.

Well, based on your performance in NEET PG 2021and previous years' cutoff marks, you have tough chance to get admission to MS in General Surgery at top-rated Government or Private medical colleges as your rank is not up to the mark.

However, you may get chance at few medical colleges in India for MS in General Surgery such as:

1. Aarupadai Veedu Medical College and Hospital, Puducherry (2019 Cutoff - 50,035 in Round 1)
2. Sree Balaji Medical College and Hospital, Chennai (2019 Cutoff - 51,903)
3. Santosh Medical College And Hospital, Santosh University (2018 Cutoff - 55,024)
4. Meenakshi Medical College Hospital and Research Institute, Chennai (2019 Cutoff - 63067 in Round 1)

Please note, this prediction has been made based on previous years' cutoff details and may be subjected to vary from actual results.

To check your chances in 50% All India, 50% State Quota Govt Seats & Private Seats, Central & Deemed Universities for your Home State, Seat type, Caste, etc. and to get a NEET PG Counseling personalised report with Top MD/ MS/ Diploma Colleges in All India & State Level Counsellings, I encourage you to use NEET PG 2021 College Predictor:

https://medicine.careers360.com/neet-pg-college-predictor?utm_source=qna&utm_medium=neet_pg_cp

I hope you find the information useful.

Thank you and all the best!