The structure of the tetraquark candidate Zc(3900), which was experimentally reported in e+e− collisions, is studied by the s-wave meson-meson coupled-channel scattering on the lattice. The s-wave interactions among the πJ/ψ, ρηc and DDˉ∗ channels are derived from (2+1)-flavor dynamical QCD simulations at mπ=410--700 MeV. It is found that the interactions are dominated by the off-diagonal πJ/ψ-DDˉ∗ and ρηc-DDˉ∗ couplings. With the interactions obtained, the s-wave two-body amplitudes and the pole position in the πJ/ψ-ρηc-DDˉ∗ coupled-channel scattering are calculated. The results show that the Zc(3900) is not a conventional resonance but a threshold cusp. A semiphenomenological analysis with the coupled-channel interaction to the experimentally observed decay mode is also presented to confirm the conclusion.