  Supermassive primordial stars forming in atomically-cooled halos at $z
\sim15-20$ are currently thought to be the progenitors of the earliest quasars
in the Universe. In this picture, the star evolves under accretion rates of
$0.1 - 1$ $M_\odot$ yr$^{-1}$ until the general relativistic instability
triggers its collapse to a black hole at masses of $\sim10^5$ $M_\odot$.
However, the ability of the accretion flow to sustain such high rates depends
crucially on the photospheric properties of the accreting star, because its
ionising radiation could reduce or even halt accretion. Here we present new
models of supermassive Population III protostars accreting at rates $0.001 -
10$ $M_\odot$ yr$^{-1}$, computed with the GENEVA stellar evolution code
including general relativistic corrections to the internal structure. We use
the polytropic stability criterion to estimate the mass at which the collapse
occurs, which has been shown to give a lower limit of the actual mass at
collapse in recent hydrodynamic simulations. We find that at accretion rates
higher than $0.001$ $M_\odot$ yr$^{-1}$ the stars evolve as red, cool
supergiants with surface temperatures below $10^4$ K towards masses $>10^5$
$M_\odot$, and become blue and hot, with surface temperatures above $10^5$ K,
only for rates $\lesssim0.001$ $M_\odot$ yr$^{-1}$. Compared to previous
studies, our results extend the range of masses and accretion rates at which
the ionising feedback remains weak, reinforcing the case for direct collapse as
the origin of the first quasars.
