The United States is the only major destination where a strong graduate applicant can plausibly be paid to study. Research and teaching assistantships routinely cover full tuition and provide a living stipend, and fellowships add to that.
The money does not come from a central scholarship office. It comes from departments, research groups, and individual faculty with grant funding, and it is allocated on their timelines and by their criteria.
Why outreach decides outcomes
Funded admission at graduate level frequently follows from a faculty member wanting a specific student. That relationship starts with contact, and effective contact demonstrates that you have read specific work, understood its direction, and have something relevant to contribute.
Mass-emailed generic enquiries are worse than no contact at all. They are visibly generic and they mark you as an applicant who did not do the reading.
Timing
US graduate cycles run early. Applications for an autumn intake commonly close in the preceding winter, and funding decisions follow shortly after. Testing, transcripts, references, and outreach therefore need to be complete in the year before you intend to start.
If the arithmetic does not work for the coming cycle, targeting the next one is a far better outcome than an underfunded, rushed application.
Building the file
A research statement that is specific about the questions you want to work on, references from people who supervised your actual work, a transcript presented clearly, and where required standardised test scores.
Undergraduate funding in the US is a different and much harder problem, with far fewer need-based options open to international students. Be realistic about which of the two you are pursuing.