Humane and Existing Alternatives in Research and Testing Sciences Act of 2021 or the HEARTS Act of 2021 This bill expands the responsibilities of the National Institutes of Health (NIH) to promote research methods that do not use animals. Specifically, the NIH must (1) provide incentives for nonanimal research, and (2) require the evaluation of viable nonanimal alternatives before approving any research that involves animals. Additionally, NIH applications for research grants involving animals must include an assurance that nonanimal methods for such research are not available.