We explore the spatial growth of disturbances developing on top of a base flow given by the Hagen–Poiseuille profile, which has been modified by a small axisymmetric and axially invariant distortion. Such deviations from the ideal parabolic profile may, for instance, occur in experiments as a result of experimental uncertainties. The optimal distortion (i.e. the distortion with a prescribed norm that induces the maximum growth rate) is computed by a variational technique. Unstable modes are found to exist for very small values of the norm of the deviation at low Reynolds numbers, and the instability is governed by an inviscid mechanism. The growth of these modes and the ensuing transition to turbulence is then studied by means of direct numerical simulations. Two possible paths of transition are found, one based on the exponential amplification of axisymmetric disturbances and the subsequent formation of $\uLambda$-vortices and the other based on the growth and breakdown of streamwise streaks.