Any bridge referred to in section 1390 may be constructed by contract, let as provided by law, by any city into which the bridge extends, or by the county into which such bridge extends or in which such bridge is located. Any such city or county may contribute toward the expense of the construction or maintenance of such bridge by the appropriation for such purpose of any money in its treasury not otherwise appropriated, upon such terms and conditions as are prescribed by ordinance or resolution of the governing body of such city.