The spreading of polymer droplets under conditions of complete wetting on an ideally flat and structureless solid substrate has been studied by computer simulation, using a coarse-grained bead-spring model of flexible macromolecules. Evidence is obtained that a power law close to Tanner's law for the growth of the lateral droplet radius {r(t)proportional tot(0.14)} and contact angle {thetaproportional tot(-0.31)} holds on nanoscopic scales. We observe the formation of a precursor film around the spreading droplet and find that the film attains diffusive dynamics at late times.