  Accreting supermassive stars of $\gtrsim 10^{5}\,M_\odot$ will eventually
collapse directly to a black hole via the general relativistic (GR)
instability. Such direct collapses of supermassive stars are thought to be a
possible formation channel for supermassive black holes at $z > 6$. In this
work, we investigate the final mass of accreting Population III stars with
constant accretion rates between $0.01$ and $1000\,M_\odot$yr$^{-1}$. We
determine the final mass by solving the differential equation for the general
relativistic linear adiabatic radial pulsations. We find that models with
accretion rates $\gtrsim 0.05\,M_\odot$yr$^{-1}$ experience the GR instability
at masses depending on the accretion rates. The critical masses are larger for
higher accretion rates, ranging from $8\times10^{4}\,M_\odot$ for
$0.05\,M_\odot$yr$^{-1}$ to $\sim10^6\,M_\odot$ for $1000\,M_\odot$yr$^{-1}$.
The $0.05\,M_\odot$yr$^{-1}$ model reaches the GR instability at the end of the
core hydrogen burning. The higher mass models with the higher accretion rates
reach the GR instability during the hydrogen burning stage.
