[en] In this paper, the lepton flavor violating τ→lP(V) (P, V=π⁰, η, η^', ρ⁰, ω, φ) and τ→3l (l=e, μ) decays are studied in the framework of the type-III seesaw model, in which new triplet fermions with a zero hypercharge (Y=0) interact with ordinary lepton doublets via Yukawa couplings, and affect tree-level leptonic Z-boson couplings. We investigate the experimental bound from the leptonic Z decay to get constraints on the existing parameters space. We predict that the upper limits on the branching ratios of τ→lP(V) and τ→3l can reach the experimental current limits.