Hi

As per previous year cutoff, you can get Mechanical Engineering at the following NITs under Home State Quota - 

National Institute of Technology, Goa

National Institute of Technology, Hamirpur

National Institute of Technology, Meghalaya

National Institute of Technology, Nagaland

National Institute of Technology, Puducherry

National Institute of Technology, Raipur

National Institute of Technology, Sikkim

National Institute of Technology, Arunachal Pradesh

National Institute of Technology, Manipur

National Institute of Technology, Mizoram

National Institute of Technology, Silchar

National Institute of Technology, Srinagar

Also, if previous year cutoff ranks are to be considered, you cannot get Mechanical Engineering at any of the NITs under Other State Quota.

Check this link to know about NITs Cutoff for B.Tech Mechanical Engineering: https://engineering.careers360.com/articles/nits-cutoff-for-btech-mechanical-engineering.

Best of Luck.

Hope this helps.

Cheers!