Hospitals may, at their election, by written agreement between the counties owning and operating the hospitals and the Illinois Department, assume specified expenses of the operation of the Illinois Department associated with the determination of eligibility, direct payment of which expenses by the Illinois Department would qualify as public funds expended by the Illinois Department for the Illinois Medical Assistance Program or other health care programs administered by the Illinois Department. The Illinois Department shall open an adequately staffed special on-site office or offices at facilities designated by the county for the purpose of assisting the county in ensuring that all eligible individuals are enrolled in the Illinois Medical Assistance Program. Each such agreement, executed in accordance with Section 3 of the Intergovernmental Cooperation Act, shall describe the operational expenses to be assumed in sufficient detail to permit the Illinois Department to certify upon such written obligation or performance thereunder that the hospital’s compliance with the terms of the agreement will amount to the commitment of public funds eligible for the federal financial participation or other federal funding called for in Title XIX or Title XXI of the Social Security Act.