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In this study, we try to extract the mass radii of the neutron and the proton from the differential cross section data of near-threshold $ω$ and $ϕ$ photoproductions on deuterium target, which is often approximated as a quasi-free neutron plus a quasi-free proton. The incoherent data of $ω$ and $ϕ$ photoproductions are provided by CBELSA/TAPS collaboration and LEPS collaboration respectively, where the deuteron is disintegrated in the experiments to measure the properties of individual nucleons. Under the VMD model and the assumption of dipole gravitational form factor, we determined the loosely bound neutron and proton mass radii to be $0.795\pm0.092\rm(stat.)\pm0.073\rm(syst.)$ fm and $0.744\pm0.029\rm(stat.)\pm0.042\rm(syst.)$ fm respectively from the near-threshold data of $γd \rightarrow ωn (p)$ and $γd \rightarrow ωp (n)$, for the first time. With the near-threshold and incoherent $ϕ$ photoproduction data of $γd \rightarrow ϕp n$, we determined the average mass radius of the bound nucleon (neutron or proton) inside the deuteron to be $0.755\pm0.039\rm(stat.)\pm0.039\rm(syst.)$ fm, for the first time. For a comparison study, we also extracted the mass radius of the free proton from the $ω$ photoproduction on the hydrogen target by CBELSA/TAPS collaboration. Based on our analysis results under the assumptions of VMD model and a low energy QCD theorem, we find that the neutron mass radius is consistent with the proton mass radius within the current statistical uncertainties, and that the nuclear modification on the nucleon mass radius is small inside the deuteron.