The estimates of expenditure requirements transmitted by the administrative head of each budgeted agency to the Secretary of the Office of Policy and Management, pursuant to section 4-77, shall include an estimate of the amount required by such agency for the payment of the workers’ compensation claims of the employees of each such agency. Appropriations which are recommended in the budget document transmitted by the Governor in the odd-numbered years or the status report transmitted by the Governor in the even-numbered years to the General Assembly pursuant to section 4-71 or contained in the state budget act or any deficiency bill, as provided in section 2-36, for the payment of such claims shall be made as follows: (1) For the Departments of Developmental Services, Mental Health and Addiction Services, Correction, Transportation, Emergency Services and Public Protection and Children and Families, directly to said agencies; (2) for all other budgeted state agencies, to the Department of Administrative Services which shall maintain an account for payment of workers’ compensation claims.