  In this exploratory study, near-threshold scattering of $D$ and $\bar{D}^*$
meson is investigated using lattice QCD with $N_f=2+1+1$ twisted mass fermion
configurations. The calculation is performed within the coupled-channel
L\"uscher's finite-size formalism. The study focuses on the channel with
$I^G(J^{PC})=1^+(1^{+-})$ where the resonance-like structure $Z_c(3900)$ was
discovered. We first identify the most relevant two channels of the problem and
the lattice study is performed within the two-channel scattering model.
Combined with a two-channel Ross-Shaw theory, scattering parameters are
extracted from the energy levels by solving the generalized eigenvalue problem.
Our results on the scattering length parameters suggest that, at the particular
lattice parameters that we studied, the best fitted parameters do not
correspond to a peak behavior in the elastic scattering cross section near the
threshold. Furthermore, within the zero-range Ross-Shaw theory, the scenario of
a narrow resonance close to the threshold is disfavored beyond $3\sigma$ level.
