Mich. Comp. Laws § 801.1
The common jails in the several counties of this state in charge of the respective sheriffs shall be used as prisons:
First, For the detention of persons charged with offenses and duly committed for trial;
Second, For the confinement of persons committed pursuant to a sentence upon conviction of an offense, and of all other persons duly committed for any cause authorized by law; and the provisions of this section shall extend to persons detained in or committed to any such jail when duly authorized by or under the authority of any court or officer of the United States, as well as by the courts and magistrates of this state: Provided, however, That all persons detained or committed to such jails by the authority of the courts of the United States, or any officer of the United States, shall be received in said county jails only in cases where the cost of the care and maintenance of such persons shall be paid by the United States, at actual cost thereof, to be fixed and determined by the Michigan welfare commission upon application of the sheriffs of the respective counties of this state, and not otherwise.
MCL 801.1