Nonlinear travelling waves that are precursors to laminar–turbulent transition and capture the main structures of the turbulent buffer layer have recently been found to exist in all the canonical parallel flow geometries. We study the effect of polymer additives on these ‘exact coherent states’ (ECS) in the plane Poiseuille geometry, focusing on Reynolds numbers slightly above transition. Many key aspects of the turbulent drag reduction phenomenon are found, including delay in transition to turbulence, drag reduction onset threshold, and diameter and concentration effects. Furthermore, examination of the ECS existence region leads to a distinct prediction, consistent with experiments, regarding the nature of the maximum drag reduction regime: at sufficiently high wall shear rates, viscoelasticity is found to completely suppress the normal (i.e. streamwise-vortex-dominated) dynamics of the near-wall region, suggesting that the maximum drag reduction regime is dominated by a distinct class of flow structures.