Close-contact melting (CCM) problems examine a heated object in contact with a solid phase-change material. Existing research determines the velocity at which the heat source melts through the solid, so that the melting process can be extrapolated to multiple operating conditions without resorting to experiments. In our contribution we present a transient CCM model for a planar heat source. Namely, by neglecting the thermo-mechanical equilibrium assumption in the melt film, we derive a time-dependent melting velocity. This new formulation results in a coupled problem, as the melting velocity depends on the evolving temperature distribution in the solid material. In this work, we describe our numerical workflow for transient CCM. After solving for the heat equation in the solid material with a space-time finite element method, we retrieve the melting velocity and employ a mesh-update method to represent the movement of the heat source. Finally, we show a two-dimensional test case and discuss extensions to realistic applications.