Sliding motion has always been one of the major concerns when it comes to the analysis of viscoelastic contact problems. A new model simulating the transient sliding contact of smooth viscoelastic surfaces is developed in this paper. By taking the dry contact friction and the coupling between shear tractions and normal pressure into account, the effect of the early partial slip period, which is often neglected in the study of viscoelastic sliding contact problems, is investigated numerically. Compared with solutions based on the frictionless assumption, the steady-state pressure profile is found to be slightly different under the effect of the partial slip regime, including a lower peak pressure and the shift of the contacting region in the direction opposite to the sliding motion. Furthermore, the time required for the viscoelastic contact to reach its steady state is delayed owing to the partial slip period preceding the global sliding motion.