It shall be the duty of the finance director to investigate and audit the claims regulated by Section 3.08.010, and recommend to the city council whether said claims should be allowed or rejected. At each regular meeting of the city council all claims having been so audited shall be allowed or disallowed by the city council, as in its judgment deems right, and the finance director shall issue warrants for the payment of such claims that have been allowed by the city council. The finance committee may audit such claims as it deems necessary on behalf of the city council.