Conway Springs is a city in Sumner County, Kansas, United States. As of the 2020 census, the population of the city was 1,086. Conway Sprin…Conway Springs is a city in Sumner County, Kansas, United States. As of the 2020 census, the population of the city was 1,086. Conway Springs was founded in 1884. Its name is derived from both Conway Township and a mineral spring nearby.