  We investigate the possibility to study the charmoniumlike state $Z_c(3900)$
through the pion-induced production off a nuclear target. By using a
high-energy pion beam, the $Z_c(3900)$ can be produced off a proton or nucleus
though the Primakoff effect. The production amplitude is calculated in an
effective Lagrangian approach combined with the vector dominance model. The
total cross sections of the $p(\pi^-, Z^-_c(3900))$ and $p(\pi^-,
Z^-_c(3900)\to J/\psi\pi^-)$ reactions are calculated, and their order of
magnitude is about 0.1 and 0.01nb, respectively, with an assumption of branch
ratio 10\% for the $Z_c(3900)$ decay in $J/\psi\pi$ channel. If the proton
target is replaced by a nuclear target, the production of the $Z_c(3900)$
enhances obviously. The predicted total cross sections for the $A(\pi^-,
Z^-_c(3900))$ and $A(\pi^-, Z^-_c(3900)\to J/\psi\pi^-)$ reactions with
$A=^{12}$C or $^{208}$Pb are on the order of magnitude of 100 and 10 nb,
respectively, which is about one thousand times larger than the cross sections
off a proton target. Based on these the results, we suggest the experimental
study of the $Z_c(3900)$ by using high-energy pion beams with a nuclear target
at facilities such as COMPASS and J-PARC.
