This study investigates the viscoelastic effects on droplet migration induced by a wettability gradient on a rigid substrate by a numerical simulation based on OpenFOAM with the volume-of-fluid method. The droplets are set with different rheological properties to investigate the effect of the elastic parameters. The Oldroyd-B model was employed. Quantitative differences in the migration and deformation between Newtonian and viscoelastic droplets were investigated by changing the degree of elasticity. The droplet migration shows conspicuously higher mobility for high elasticity, especially during the accelerating period. Moreover, the displacement and velocity increased with the decrease of a viscoelasticity parameter, and the velocity enhancement was regulated by the elastic instability shown at a high Weissenberg number. In addition, the velocity of the droplet changes more significantly over the range of contact angles of 130 degrees to 60 degrees compared to other wettability-gradient surfaces.