Washington is a city in Daviess County, Indiana. The population was 12,017 at the time of the 2020 census. The city is the county seat of …Washington is a city in Daviess County, Indiana. The population was 12,017 at the time of the 2020 census. The city is the county seat of Daviess County. It is also the principal city of the Washington, Indiana Micropolitan Statistical Area, which comprises all of Daviess County and had an estimated 2017 population of 31,648.