We present results of a study of the decay J/ψ → ωηπ+ π- using a sample of (225.2 ± 2.8) × 10(6) J/ψ events collected by the BESIII detector, and report the observation of a new process J/ψ → ωX(1870) with a statistical significance of 7.2σ, in which X(1870) decays to a(0)(±)(980)π±. Fitting to ηπ+ π- mass spectrum yields a mass M = 1877.3 ± 6.3(stat)(-7.4)(+3.4)(syst) MeV/c(2), a width Γ = 57 ± 12(stat)(-4)(+19)(syst) MeV/c(2), and a product branching fraction B(J/ψ → ωX) × B(X→a(0)(±)(980)π±) × B(a(0) (±)(980) → ηπ±) = [1.50 ± 0.26(stat)(-0.36)(+0.72) (syst)] × 10(-4). Signals for J/ψ → ωf(1)(1285) and J/ψ → ω η(1405) are also clearly observed and measured.