The city council may, at any time within 20 days after the opening of bids, or, if no bids are received, at any time within 20 days after the time fixed for the opening of bids, order by resolution adopted by a vote of two-thirds of all of its members, that the city itself execute the work of the improvement, or any part thereof, in accordance with the plans and specifications adopted for such work, or for such part thereof, and employ the labor and provide the materials, appliances, supplies or illuminating agent necessary therefor. The cost and expenses of such work, or of such part thereof, shall be paid out of the special fund collected to pay for the improvement. If the city council orders that only a part of the work of the improvement be executed by the city then the city council may reject all bids, if any, received for such part of the work and may proceed with the award of the contract for the remaining part of the work.