  We study the behaviour of the $\chi_{c1}(3872)$, also known as $X(3872)$, in
dense nuclear matter. We begin from a picture in vacuum of the $X(3872)$ as a
purely molecular $(D \bar D^*-c.c.)$ state, generated as a bound state from a
heavy-quark symmetry leading-order interaction between the charmed mesons, and
analyze the $D \bar D^*$ scattering $T-$matrix ($T_{D \bar D^*}$) inside of the
medium. Next, we consider also mixed-molecular scenarios and, in all cases, we
determine the corresponding $X(3872)$ spectral function and the $D \bar D^*$
amplitude, with the mesons embedded in the dense environment. We find important
nuclear corrections for $T_{D \bar D^*}$ and the pole position of the
resonance, and discuss the dependence of these results on the $D \bar D^*$
molecular component in the $X(3872)$ wave-function. These predictions could be
tested in the finite-density regime that can be accessed in the future CBM and
PANDA experiments at FAIR.
