In this Rapid Communication, we theoretically demonstrate that near-field radiative heat transfer (NFRHT) can be modulated and enhanced by a new energy transmission mode of evanescent wave, i.e. the nonreciprocal surface plasmons polaritons (NSPPs). In addition to the well-known coupled surface plasmon polaritons (SPPs), applying a drift current on a graphene sheet leads to an extremely asymmetric photonic transmission model, which has never been noted in the noncontact heat exchanges at nanoscale before. The coupling of plasmons in the infrared bands dominates the NFRHT, associated with low loss (high loss and ultrahigh confinement) traveling along (against) the current. The dependence of NSPPs on the drift-current velocity as well as the vacuum gap is analyzed. It is found that the coupling of NSPPs at smaller and larger gap sizes exhibits different nonreciprocities. Finally, we also demonstrate that the prominent influence of the drift current on the radiative heat flux is found at a low chemical potential. These findings will open a new way to spectrally control NFRHT, which holds great potential for improving the performance of energy systems like near-field thermophotovoltaics and thermal modulator.