Can Galileons support Lorentzian wormholes?

We discuss the possibility of constructing stable, static, spherically symmetric, asymptotically flat Lorentzian wormhole solutions in General Relativity coupled to a generalized Galileon field $\pi$. Assuming that Minkowski space-time is obtained at $\mathrm{\partial }\pi =0$, we find that there is tension between the properties of the energy-momentum tensor required to support a wormhole (violation of average null energy conditions) and stability of the Galileon perturbations about the putative solution (absence of ghosts and gradient instabilities). In 3-dimensional space-time, this tension is strong enough to rule out wormholes with above properties. In higher dimensions, including the most physically interesting case of 4-dimensional space-time, wormholes, if any, must have fairly contrived shapes.