Grant Integrity and Border Security Act This bill requires an applicant for a federal grant to certify that the applicant has not violated in the last 10 years, is not currently violating, and will not violate during the term of the grant the federal criminal prohibition against bringing into the United States or harboring certain non-U.S. nationals (i.e., aliens under federal law). Upon determination by the granting agency that a grantee is in violation of this bill, the agency must withhold any federal grant funds from the grantee. The Department of Justice (DOJ) must report to the Office of Management and Budget (OMB) individuals convicted of, or who admitted to, such a violation. Additionally, the DOJ must report to OMB if DOJ believes there is a reasonable basis to conclude that an individual has committed such a violation.