Linn is a city in Osage County, Missouri, United States. The population was 1,350 at the 2020 census. It is the county seat of Osage County…Linn is a city in Osage County, Missouri, United States. The population was 1,350 at the 2020 census. It is the county seat of Osage County. Linn is part of the Jefferson City, Missouri Metropolitan Statistical Area.