  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.
