Hey so here's the deal. If you get into MBB for consulting, sponsorship will not hold you back. Actually not sure about Bain, but BCG and McKinsey won't even think twice about sponsoring your visa if they think you're worth it. Make sure you do all kinds of prep from day one because it's very competitive.
Outside of that, Deloitte advisory also sponsors visas. Deloitte Consulting does not. Smaller firms won't even look at your resume (it will never even be read by HR if you answer 'yes' to the 'do you need sponsorship' question which is present in all applications). You can get lucky with managerial roles in random companies but that's based on luck more than anything else. Overall, I'd say don't waste your time going to the US if you're Indian (your username suggests you are). EU provides pathways to residency. The US does not, and you'll be a slave to your H1B for your entire working life (unless you marry an american or wait 21 years after having a kid in the US.)
Also, the 'goodness' of an MBA won't actually matter if you're targeting MBB. As long as they do campus recruitment at your university, you will be interviewed. Typically expect to stay in the same state as your university. A top tier MBA will give you more options to work in a different MBB office than your University's state. And as a ground rule, if you get a summer internship at one of the companies, they will extend a full time offer. If you don't do a summer internship at those companies, you will have a much harder time getting in.