Given all that
@FillyofDelphi pointed out here, it might be more likely to see colonies developed from individual, fragmented states within the Indian subcontinent as opposed to explicitly "Indian" colonies. And then, eventually, those fragmented states could be united under a single power which takes control of the colonies, and perhaps expands and/or adds to them. Still though, what motivates those states to colonize, I'm not totally certain.
Edit: Given an
extremely cursory knowledge of Indian history, I'd say that the Vijayanagara Empire strikes me as your best bet. Apparently religiously tolerant, as well as having an economy that was seemingly very important for the times. According to Wikipedia, "Large vessels from China made frequent visits, some captained by the Chinese Admiral Zheng He, and brought Chinese products to the empire's 300 ports, large and small, on the Arabian Sea and the Bay of Bengal."
If you can give the Vijayanagra an incentive to build a strong navy, I don't see why they wouldn't be interested in establishing some trade outposts for access to additional goods. Those outposts could then become colonies.