  The decays $\tau \rightarrow (\eta,\eta') K^{-} \nu_{\tau}$ are described in
the framework of the extended Nambu-Jona-Lasinio model. Both full and
differential widths of these decays are calculated. The vector and scalar
channels are considered. In the vector channels, the subprocesses with
intermediate $K^{*}(892)$ and $K^{*}(1410)$ mesons play the main role. In the
scalar channels, the subprocesses with intermediate $K_{0}^{*}(800)$ and
$K_{0}^{*}(1430)$ mesons are taken into account. Scalar channels play a less
important role in calculation of the full width of the decay $\tau \rightarrow
(\eta,\eta') K^{-} \nu_{\tau}$. The obtained results are in satisfactory
agreement with the experimental data.
