  The study of medium effects on properties of particles embedded in nuclear
matter is of great importance for understanding the nature and internal
quark-gluon organization as well as exact determination of the quantum numbers,
especially of the exotic states. In this context, we study the physical
properties of one of the famous charmonium-like states, $Z_c(3900)$, in a cold
dense matter. We investigate the possible shifts in the mass and current-meson
coupling of the $Z_c(3900)$ state due to the dense medium at saturation
density, $ \rho^{sat} $, by means of the in-medium sum rules. We also estimate
the vector self-energy of this state at saturation nuclear matter density. We
discuss the behavior of the spectroscopic parameters of this state with respect
to the density up to a high density corresponding to the core of neutron stars,
$\rho\approx 5\rho^{sat}$. Both the mass and current-coupling of this state
show nonlinear behavior and decrease with respect to the density of the medium:
the mass reaches roughly $30\%$ of its vacuum value at $ \rho=5\rho^{sat} $,
while the current-coupling approaches zero at $ \rho\approx2.1\rho^{sat} $,
when the central values of the auxiliary and other input parameters are used.
