I agree with @TizerorFizz and think the Fulbright Foundation will be an excellent starting point.
I got my PhD in America and taught in two excellent American unis for about 15 years. I am writing now to bring up a possibility we don’t have in the UK, where you can also find excellent financial aid: the elite four year liberal arts colleges. I am writing from an airport, feeling distracted, so this is rushed. These are called colleges because they don’t offer doctorates or, sometimes, any postgraduate degrees. The academic staff are superbly qualified. They have chosen to focus on teaching rather than research ( though many do some research). Students have much greater interaction with professors than at universities, particularly in the early years.
One of DH’s best ever PhD students, now a professor at one of the most prestigious Russell Group unis, attended such a college for UG. Preparation for PG study and employment is superb. A mum on this Board recently wrote that she had a DC at Oxbridge, a DC at an Ivy, and a DC at one of these colleges. The college DC had by far the best experience.
I have never had a connection with such a college but I observed that alumni were usually very well prepared for PG studies and had high completion rates.
Some of the elite four year colleges also offer generous aid to international students.
Examples of such colleges - though only Amherst sticks in my mind regarding aid to international students - include but are by no means limited to Amherst, Oberlin, Reed, Haverford (? - Bryn Mawr’s brother for men), Bowdoin, Vassar, Scripps/Pomona/Claremont/Harvey Mudd/Pitzter (sp?) - this is a consortium of top colleges just east of LA. Different emphases and they co-operate - then also a good dozen others of roughly the same calibre I can’t think of at the moment.
Students and alumni are also very plugged into employment and professional networks; name recognition for the best colleges is right up there with the best universities. Worth thinking about, IMO.