Morgan is a city in Bosque County in Central Texas, United States. The city's population was 454 at the 2020 census. According to the Uniteā¦Morgan is a city in Bosque County in Central Texas, United States. The city's population was 454 at the 2020 census. According to the United States Census Bureau, the city has a total area of 0.8 square miles, all land.