I am a CS major at Courant at NYU. Whatever I say here is from my experience in the CS PhD program, and might be extrapolated to math as well.

My suggestion is that you could consider applying for PhD programs directly in the US, without a masters, since many PhD programs in the US, to my knowledge, do not require a masters degree. The PhD programs here are usually self-contained, with certain mandatory course requirements and freedom to take additional courses of your choice so that you learn many advanced concepts before conducting research. Plus, PhD students here at Courant are guaranteed funding for the first five years of your program, which is the typical time taken to complete the program or get close to completion. Many universities here also have combined and fully-funded MS+PhD programs.

I was in a similar situation as you are in now, when I was finishing up my B.Tech in CS in India. I was confused among three options - masters in India, masters in US, PhD in US. I ended up working in industry for couple of years in India to help make a decision more slowly, and this is what I learned. Masters in the US is useful mainly only in two situations - i) you are changing fields, and you wish to show strong coursework in your new field in a reputed college or university, ii) you are not interested in PhD and just want a high-paying job to settle. Masters in India is a very good option, especially in good universities such as the IITs, and there is no doubt that you will certainly increase your chances of securing admission in a good university in the US. A masters degree conveys the message of greater conviction and dedication to the field of your interest. With good recos and good grades, and maybe some publications (depending on your field), you are certain to secure a very good spot in the US.

But if you i) already have good grades, ii) aren't changing your fields, iii) already are clear in your areas of interest and the type of research problems you want to work on, then my advice is to apply for PhD or MS+PhD programs in the US, which are fully funded in most cases and also have coursework.