A PDE-based model combining surface electromigration and wetting is developed for the analysis of the morphological stability of ultrathin solid films. The adatom mobility is assumed anisotropic, and two directions of the electric field (parallel and perpendicular to the surface) are discussed and contrasted. The linear stability analysis of the small-slope evolution equations is followed by computations of the fully nonlinear parametric evolution equations that permit surface overhangs. The results reveal parameter domains of instability for wetting and non-wetting films and variable electric field strength, nonlinear steady-state solutions in certain cases, and interesting coarsening behavior for strongly wetting films. The goal of the modeling at this stage is not to match theoretical results with experiments, but rather to provide a broad analysis of the interplay of two effects (wetting and electromigration) that, at least to our knowledge, has not been addressed in prior publications.