We investigate a model which couples diffusional melting and nanoscale structural forces via a combined nano-mesoscale description. Specifically, we obtain analytic and numerical solutions for melting processes at grain boundaries influenced by structural disjoining forces in the experimentally relevant regime of small deviations from the melting temperature. Though spatially limited to the close vicinity of the tip of the propagating melt finger, the influence of the disjoining forces is remarkable and leads to a strong modification of the penetration velocity. The problem is represented in terms of a sharp interface model to capture the wide range of relevant length scales, predicting the growth velocity and the length scale describing the pattern, depending on temperature, grain boundary energy, strength and length scale of the exponential decay of the disjoining potential. Close to equilibrium the short-range effects near the triple junctions can be expressed through a contact angle renormalisation in a mesoscale formulation. For higher driving forces strong deviations are found, leading to a significantly higher melting velocity than predicted from a purely mesoscopic description.