The present study investigates the modal stability of the steady incompressible flow inside a toroidal pipe for values of the curvature (ratio between pipe and torus radii) approaching zero, i.e. the limit of a straight pipe. The global neutral stability curve for is traced using a continuation algorithm. Two different families of unstable eigenmodes are identified. For curvatures below, the critical Reynolds number is proportional to. Hence, the critical Dean number is constant,. This behaviour confirms that the Hagen-Poiseuille flow is stable to infinitesimal perturbations for any Reynolds number and suggests that a continuous transition from the curved to the straight pipe takes place as far as it regards the stability properties. For low values of the curvature, an approximate self-similar solution for the steady base flow can be obtained at a fixed Dean number. Exploiting the proposed semi-analytic scaling in the stability analysis provides satisfactory results.