When it becomes necessary in the prosecution of any work or improvement done under contract to make alterations or modifications in such contract, such alterations or modifications shall be made only upon resolution of the Council. No order for such alterations or modifications shall be effective until the price to be paid for the material and work, or both, under the altered or modified contract shall have been agreed upon in writing and signed by the contractor and the City Manager upon authority of the Council, and a copy thereof and of the proceedings authorizing such alteration or modification certified by the Clerk, and attached by him to the original contract on file in his office.