We study the $J/\psi \pi$ and $D{\overline{D}}^{*}$ coupled-channel system within a covariant framework. The $J/\psi \pi$ and $D{\overline{D}}^{*}$ invariant-mass distributions measured at 4.23 and 4.26 GeV by BESIII and the finite-volume energy levels from recent lattice QCD simulations are simultaneously fitted. Phase shifts and inelasticities of the $J/\psi \pi$ and $D{\overline{D}}^{*}$ scattering are predicted using the resulting amplitudes. Poles corresponding to the $Z_c\left(3900\right)$ state are found in the complex energy plane and their couplings with $J/\psi \pi$ and $D{\overline{D}}^{*}$ are determined. Our results indicate that the current lattice data do not preclude the existence of the $Z_c\left(3900\right)$ state.