In any district which has, or any special district which has, a governing board composed of elective members, except the governing board of any school district, if such district is not required by any other provision of law to publish notice of election of members to its governing board, the governing board shall cause a notice of such election to be published in a newspaper of general circulation at least 10 days before the time of filing nominations for office in the manner prescribed by Section 6061 of this code. If there is no newspaper of general circulation within the district, the governing board shall cause a notice of such election to be posted in three prominent public places within the district at least 10 days before the time of filing nominations for office.