  In the presented paper production of charmonium state $\mathcal{Q}$ in
exclusive $W\to\mathcal{Q} D_{s}^{(*)}$ decays is analyzed in the framework of
both leading order Nonrelativistic Quantum Chromodynamics (NRQCD) and
light-cone expansion (LC) models. Analytical and numerical predictions for the
branching fractions of these decays in both approaches are given. The typical
value of the branching fractions is $\sim 10^{-11}$ and it turns out the LC
results are about 4 times lager than NRQCD ones, so the effect of internal
quark should be taken into account. Some estimates of color-octet contributions
are presented and it is shown, that these contributions could be comparable
with color-singlet results.
