Wormholes supported by the kink-like configuration of a scalar field

We study the problem of existence of static spherically symmetric wormholes supported by the kink-like configuration of a scalar field. With this aim we consider a self-consistent, real, nonlinear, nonminimally coupled scalar field $\phi$ in general relativity with the symmetry-breaking potential $V(\phi)$ possessing two minima. We classify all possible field configurations ruling out those of them for which wormhole solutions are impossible. Field configurations admitting wormholes are investigated numerically. Such the configurations represent a spherical domain wall localized near the wormhole throat.