  We study the phenomenology of the ground-state (pseudo-)scalar charmonia
$\eta_c$ and $\chi_{c0}$ in the framework of a $U(4)_r \times U(4)_l$ symmetric
linear sigma model with (pseudo-)scalar and (axial-) vector mesons. Based on
previous results for the spectrum of charmonia and the spectrum and
(OZI-dominant) strong decays of open charmed mesons, we extend the study of
this model to OZI-suppressed charmonia decays. This includes decays into
'ordinary' mesons but also particularly interesting channels with
scalar-isoscalar resonances $f_0(1370),\, f_0(1500),\, f_0(1710)$ that may
include sizeable contributions from a scalar glueball. We study the variation
of the corresponding decay widths assuming different mixings between glueball
and quark-antiquark states. We also compute the decay width of the pseudoscalar
$\eta_c$ into a pseudoscalar glueball. In general, our results for decay widths
are in reasonable agreement with experimental data where available. Order of
magnitude predictions for as yet unmeasured states and channels are potentially
interesting for BESIII, Belle II, LHCb as well as the future PANDA experiment
at the FAIR facility.
