The present study is concerned with exploring the ability of eddy-viscosity based RANS models to compute conjugate heat transfer problems, including capturing the decay of turbulent temperature fluctuations across the wall region. This is done by extending the application of the modelled transport equations for temperature variance and its dissipation rate across the solid wall region, thus providing the decay of the variance across this region. Comparisons are drawn with existing DNS data of plane channel flow, at a relatively low Reynolds number, heated through a wall of non-zero thickness. Cases are simulated covering a range of thermal activity ratios, representing a range of different solid/fluid thermal conductivity/diffusivity ratios. It is found that, in order to predict the correct decay of temperature variance across the wall, an accurate representation of it, and of its dissipation rate, in the near-wall fluid flow region is required. This, in turn, also requires an accurate representation of the near-wall dynamic field turbulence. A number of modifications are proposed to an existing four-equation k--2- model in order to provide good near-wall predictions of the four variables under the two limiting thermal boundary treatments of isothermal and iso-flux conditions. The resulting scheme is shown to perform well in the conjugate calculations over a range of thermal activity ratios.