National Institutes of Health (NIH) is part of the U.S. Department of Health and Human Services and the primary federal grantmaker for health-related scientific research. Foreign universities can apply to NIH either as the main applicant or as a partner, but in general NIH only makes grants to foreign institutions that offer qualifications which cannot be found equivalently in the U.S.
National Science Foundation (NSF) supports basic research within areas other than the medical. Foreign research institutions are able to participate as partners in projects supported by NSF, but cannot be main applicants. The Research Support Office has registered AU in NSF’s application facility “Fastlane”. Thus, before you can participate in an application, the Research Support Office must create a “Fastlane”-profile for you.
Besides the federal funders there are also private American foundations which make grants to organizations in other countries (e.g., the Bill and Melinda Gates Foundation, the Michael J. Fox Foundation for Parkinson’s Research, etc.).