Experimental investigations of the head-on quenching of a laminar methane flame have produced conflicting statements about the magnitude of the wall heat flux during quenching and its trend with respect to wall temperature. The current theoretical formulations fail to predict the correct behavior. We have been studying the head-on quenching of a laminar, stoichiometric methane flame at atmospheric pressure in a range of wall temperatures between 300 K and 600 K using numerical simulation. To this end we solved the fully compressible, one-dimensional Navier-Stokes equations with detailed mechanisms for kinetics and diffusion (including cross-transport effects, i.e., Soret and Dufour effect). Four different chemical schemes (two where only the C-1 path is included and two that contain also the C-2 path) were used in order to minimize uncertainties resulting from different descriptions of the chemical kinetics and to investigate the influence of the C-2 chemistry. The wall is considered as chemically inert. Points of interest were the variation of the wall heat flux with wall temperature, as well as the time evaluation of species mass fractions, net heat release rates per species, and detailed reaction rates during quenching at wall temperatures of 300 K and 600 K. The computation results show an increasing dimensional wall heat flux with increasing wall temperature. The calculated wall heat flux matches the most recent experimental data up to a wall temperature of 400 K, but shows an increasing discrepancy in its actual values with increasing wall temperature. This is due to a dramatically increasing radical concentration (H, O, OH) at the wall with increasing wall temperatures and subsequent strongly exothermic radical recombination reactions. Because of the extremely low radical concentration at the wall in the low wall temperature regime (300 < T-W < 400 K), the wall can be modeled here as chemically inert and thermal diffusion processes play a negligible role during quenching. However, this is not the case for higher wall temperatures. Furthermore, those high radical concentrations at the wall in this case indicate that the actual uncertainties of wall heat flux measurements at high wall temperatures might be underestimated by experimentalists.