The commissioner of corrections may transfer an inmate of any state correctional facility to a state institution under the control of the commissioner of human services or to a private medical facility for diagnosis, treatment, or care which is not available at any state correctional facility and shall cause a proper record to be made at the institutions or facility to which a transfer has been made and at the commissioner’s office. No transfer shall be made by the commissioner of corrections without the approval of the commissioner of human services or the chief executive officer of the private facility. An inmate of any state correctional facility so transferred shall be returned to the correctional facility from which transferred by order of the commissioner of corrections upon conclusion of treatment, or, if the inmate becomes eligible for release from custody pursuant to the terms of the sentence prior to conclusion of treatment, the inmate shall be released unless, before conclusion of treatment, the inmate has been committed to a medical institution by competent authority as provided by law. The superintendent of any state institution or the chief executive officer of any private facility shall at once notify the commissioner of corrections if there is any question as to the propriety of the commitment or detention of any inmate admitted to their institution or facility and the commissioner shall immediately take action on the question.