We report a measurement of the νμ charged current quasielastic cross-sections on carbon in the T2K on-axis neutrino beam. The measured charged current quasielastic cross-sections on carbon at mean neutrino energies of 1.94 GeV and 0.93 GeV are (11.95±0.19(stat)+1.82−1.47(syst))×10−39  cm2/neutron, and (10.64±0.37(stat)+2.03−1.65(syst))×10−39  cm2/neutron, respectively. These results agree well with the predictions of neutrino interaction models. In addition, we investigated the effects of the nuclear model and the multi-nucleon interaction.