Linn is a city and the county seat of Osage County, Missouri, United States. The population was 1,350 at the 2020 census. Linn is part of t…Linn is a city and the county seat of Osage County, Missouri, United States. The population was 1,350 at the 2020 census. Linn is part of the Jefferson City, Missouri Metropolitan Statistical Area. Linn was platted in 1843. The community was named for Senator Lewis F. Linn. A post office called Linn has been in operation since 1844.