This paper deals with the stability analysis of an axially symmetric liquid metal flow driven by a rotating magnetic field in a cylinder of finite dimensions. The limit of linear stability with respect to axially symmetric perturbations is found for diameter-to-height ratios between 0.4 and 1. This oscillatory instability is shown to be different from the expected Taylor–Görtler vortices. Several linearly unstable steady solutions are found close to the stable basic state. It is shown that small finite-amplitude perturbations in the form of Taylor–Görtler vortices give rise to instability in the linearly stable regime.