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The on-shell properties of the nucleon effective mass in asymmetric nuclear matter are investigated in the framework of an extended Brueckner-Hartree-Fock (BHF) approach.The proton and neutron effective masses in neutron-rich nuclear matter are predicted by including both the effect of ground state correlations and the threebody force (TBF) rearrangement contribution.Within this framework,the neutron effective mass is predicted *to be larger than the proton one in neutron-rich nuclear matter,i.e.,m*n ≥ m*p.The effect of ground state correlations tus out to be dominated at low densities and it leads to a strong enhancement of the effective mass.The TBF rearrangement contribution becomes predominant over the effect of ground state correlations at high densities and it reduces remarkably the absolute magnitude of the isospin splitting of the neutron and proton effective masses in neutron-rich matter at high densities.