Notwithstanding the provisions of § 15.2-1609.5, the governing body of any county or city may, with the approval of the Compensation Board, enter into such agreement with the sheriff of such county or city with respect to the traveling expenses, including the use of privately owned vehicles, of such sheriff and his deputies as the governing body may deem proper. With the consent of the Compensation Board, in any county having a regular police force authorized by law and in which the jail of another county or city has been adopted as the jail of such county, the police officers, in place of the sheriff, who transport any persons charged with violation of a state law under order of the judge of the circuit court of such county to the jail so adopted, shall receive the same mileage as the sheriff would have received had he transported such persons. Any such police officer transporting any such person shall make claim for mileage on the same forms the sheriff uses for such claims and in the same manner. When any such mileage is collected by any police officer, he shall pay the same into the county treasury and the payment of such mileage shall be made in the manner provided for the payment of mileage to sheriffs.