Albany is a home rule-class city in Clinton County, Kentucky, in the United States. The population was 1,760 as of the 2020 census. It is the cou…Albany is a home rule-class city in Clinton County, Kentucky, in the United States. The population was 1,760 as of the 2020 census. It is the county seat of Clinton County. It is located on U.S. Route 127, about 6 miles north of the Tennessee border.