The ability of soft, coarse-grained models to describe the narrow interface of a nearly incompressible polymer melt in contact with a solid is explored by numerical self-consistent field calculations and Monte-Carlo simulations. We investigate the effect of the discreteness of the bead-spring architecture by quantitatively comparing the results of a bead-spring model with different number of beads, N, but identical end-to-end distance, Re, and a continuous Gaussian-thread model. If the width, ξ, of the narrow polymer–solid contact is smaller or comparable to the length of a statistical segment, , strong differences in the interface tension and the density profiles between the two models are observed, and strategies for compensating the discrete nature of the bead-spring model are investigated. Compensating the discretization of the chain contour in the bead-spring model by applying an external segment–solid potential, we simultaneously adjust the interface tension and the density profile to the predictions of the Gaussian-thread model. We suggest that the geometry of the polymer–solid contact and the interface tension are relevant characteristics that a coarse-grained model of polymer–solid contacts must reproduce in order to establish a quantitative relationship to an experimental system.