We consider the problem of the influence of a toroidal magnetization on a cylindrical surface plasmon polariton (SPP) propagating along a nanowire of a circular cross section. It follows from the dispersion equations that the SPP wavenumber linearly depends on the toroidal moment and the effect of magneto-optical nonreciprocity appears. The numerical solution of the dispersion equations demonstrates that the corresponding splitting of the SPP dispersion curves for two opposite directions of the toroidal moment is increased by an order of magnitude with respect to the planar case. The largest values of this splitting are observed for systems with relatively low optical losses, as is demonstrated by calculations for SPPs in a gold cylinder surrounded by rare-earth bismuth iron garnet.