The wetting kinetics of a water nanodroplet undergoing evaporation on a heated gold substrate were examined using molecular dynamics (MD) simulations. Various substrate and initial droplet temperatures were used to obtain different evaporation rates. The water molecule absorption-desorption behavior was analyzed in the vicinity of the contact line region to show the microscopic details of the spreading-evaporating droplet. Increasing substrate temperatures greatly affected the dynamic wetting process, while the initial water droplet temperature had very little effect. The effects of droplet size and substrate wettability on the droplet spreading-evaporating process were also examined. The radius versus time curves agree well with molecular kinetics theory (MKT) for spreading without evaporation but differ from MKT when the spreading induced evaporation. The enhancement of the wetting kinetics by the evaporation can be attributed to the reduction of the liquid-vapor surface tension and the increased water molecule motion in the contact line region and in the bulk droplet.