Recent measurements of semileptonic B-meson decays into τ leptons are somewhat higher than expected in the standard model. Although the deviations are less than 3σ, they suggest the possibility of new physics affecting primarily the τ lepton. In this paper we examine these results within the context of nonuniversal left-right models. We find that strong constraints from b→sγ on W-W′ mixing lead to a prediction of approximately equal enhancements for the B→Dτν and the B→D â †τν modes. The model predicts approximately the same enhancement for the inclusive semileptonic rate B̄→X cτν as well as for the leptonic decay Bc-→τ -ν. An enhancement in the leptonic decay B -→τ-ν is also possible but is not uniquely correlated with the other modes. For this explanation to be viable, the mass of the nonuniversal W′ would be below 1 TeV, within LHC reach.