We study the motion of magnetic skyrmions in a nanowire induced by a spin-wave current J flowing out of a driving layer close to the edge of the wire. By applying micromagnetic simulation and an analysis of the effective Thiele equation, we find that the skyrmion trajectory is governed by an interplay of both forces due to the magnon current and the wire boundary. The skyrmion is attracted to the driving layer and is accelerated by the repulsive force due to the wire boundary. We consider both cases of longitudinal and transverse driving to the nanowire, but a steady-state motion of the skyrmion is only obtained for a transverse magnon current. For the latter case, we find in the limit of low current densities J the velocity-current relation where v is the skyrmion velocity and α is the Gilbert damping. For large J, in case of strong driving, the skyrmion is pushed into the driving layer, resulting in a drop in skyrmion velocity and, eventually, the destruction of the skyrmion.