The first evidence for the decay B^{-}→D^{**0}τ^{-}ν[over ¯]{τ} is obtained using proton-proton collision data collected by the LHCb experiment, corresponding to an integrated luminosity of 9 fb^{-1}, at centre-of-mass energies of 7, 8, and 13 TeV. Here, the D^{**0} meson represents any of the three excited charm mesons D{1}(2420)^{0}, D_{2}^{}(2460)^{0}, and D_{1}^{'}(2400)^{0}. The B^{-}→D^{**0}τ^{-}ν[over ¯]{τ} signal is measured with a significance of 3.5σ, including systematic uncertainties. The combined branching fraction B(B^{-}→D{1,2}^{**0}τ^{-}ν[over ¯]{τ})×B(D{1,2}^{**0}→D^{+}π^{-}), where D_{1,2}^{**0} denotes both D_{1}(2420)^{0} and D_{2}^{}(2460)^{0} contributions, is measured to be [0.051±0.013(stat)±0.006(syst)±0.009(ext)]%, where the last uncertainty reflects that of the branching fraction of the normalization channel B^{-}→D_{1,2}^{**0}D_{s}^{-}^{()}. The ratio between the tauonic and muonic semileptonic B decays, with the latter taken from world average values, is also determined and found to be R(D_{1,2}^{**0})=0.13±0.03(stat)±0.01(syst)±0.02(ext).