Tropical mosquito-borne viruses have been expanding into more temperate regions in recent decades. This is partly due to the coupled effects of temperature on mosquito life history traits and viral infection dynamics and warming surface temperatures, resulting in more suitable conditions for vectors and virus transmission. In this study, we use a deterministic ordinary differential equations model to investigate how seasonal and diurnal temperature fluctuations affect the potential for dengue transmission in six U.S. cities. We specifically consider temperature-dependent mosquito larval development, adult mosquito mortality, and the extrinsic incubation period of the virus. We show that the ability of introductions to lead to outbreaks depends upon the relationship between a city's temperature profile and the time of year at which the initial case is introduced. We also investigate how the potential for outbreaks changes with predicted future increases in mean temperatures due to climate change. We find that climate change will likely lead to increases in suitability for dengue transmission and will increase the periods of the year in which introductions may lead to outbreaks, particularly in cities that typically have mild winters and warm summers, such as New Orleans, Louisiana, and El Paso, Texas. We discuss our results in the context of temperature heterogeneity within and across cities and how these differences may impact the potential for dengue emergence given present day and predicted future temperatures.