  We study the static and spherically symmetric wormhole spacetime
configurations at present time in a viscous universe. Considering three classes
of viscous models, i.e., bulk viscosity as a function of Hubble constant $H_0$,
present cosmic temperature $T_0$ and dark energy density $\rho_0$,
respectively, we obtain several wormhole solutions. Through the analysis for
the anisotropic solutions, we conclude that, to some extent, these three
classes of viscous models have very high degeneracy with each other, and
deviate slightly from the ideal fluid consequence. Subsequently, without the
loss of generality, to investigate the traversabilities, energy conditions and
stability for the wormhole solution, we study the wormhole solution of the
constant redshift function of the viscous $\omega$CDM model with a constant
bulk viscosity coefficient. We obtain the following conclusions: the value of
traversal velocity decreases for decreasing bulk viscosity, and the traversal
velocity for a traveler depends on not only the wormhole geometry but also the
effects of cosmological background evolution; the null energy condition will be
violated more clearly when the bulk viscosity or the cosmic expansion velocity
decreases; for the case of a positive surface energy density, the range of the
junction radius decreases and the values of the parameter $\lambda$ are further
restricted when the throat radius of the wormhole $r_0/M$ increases, bulk
viscosity coefficient $\zeta_0$ decreases and present-day Hubble parameter
$H_0$ decreases, respectively; for the case of a negative surface energy
density, by increasing $r_0/M$, decreasing $\zeta_0$ and increasing $H_0$, the
values of the parameter $\lambda$ are less restricted, one may conclude that
the total stability region increases, and the range of the junction radius
increases, increases and decreases, respectively. \end{abstract}
