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Considering the constraints of the lepton flavor violating (LFV) processes μ→3e and τ→3μ on the LFV couplings Z′ℓiℓj, in the contexts of the E6 models, the left–right (LR) models, the “alternative” left–right (ALR) models and the 331 models, we investigate the contributions of the extra gauge boson Z′ to the decay rates of the processes ℓi→ℓjνℓνℓ, τ→μP and P→μe with P=π0,η and η′. Our numerical results show that the maximal values of the branching ratios for these processes are not dependent on the Z′ mass MZ′ at leader order. The extra gauge boson ZX′ predicted by the E6 models can make the maximum value of the branching ratio Br(τ→μνℓνℓ) reach 1.1×10−7. All Z′ models considered in this paper can produce significant contributions to the process τ→μP. However, the value of Br(P→μe) is far below its corresponding experimental upper bound.