  Supermassive black holes are a key ingredient of galaxy evolution. However,
their origin is still highly debated. In one of the leading formation
scenarios, a black hole of $\sim100$ M$_{\odot}$ results from the collapse of
the inner core of a supermassive star ($\gtrsim 10^{4-5}$ M$_{\odot}$), created
by the rapid accumulation ($\gtrsim 0.1 $ M$_{\odot}$ yr$^{-1}$) of pristine
gas at the centre of newly formed galaxies at $z\sim 15$. The subsequent
evolution is still speculative: the remaining gas in the supermassive star can
either directly plunge into the nascent black hole, or part of it can form a
central accretion disc, whose luminosity sustains a surrounding, massive, and
nearly hydrostatic envelope (a system called a "quasi-star"). To address this
point, we consider the effect of rotation on a quasi-star, as angular momentum
is inevitably transported towards the galactic nucleus by the accumulating gas.
Using a model for the internal redistribution of angular momentum that
qualitative matches results from simulations of rotating convective stellar
envelopes, we show that quasi-stars with an envelope mass greater than a few
$10^{5}$ M$_{\odot} \times (\rm black~hole~mass/100 M_{\odot})^{0.82}$ have
highly sub-keplerian gas motion in their core, preventing gas circularisation
outside the black hole's horizon. Less massive quasi-stars could form but last
for only $\lesssim 10^4$ years before the accretion luminosity unbinds the
envelope, suppressing the black hole growth. We speculate that this might
eventually lead to a dual black hole seed population: (i) massive ($>10^{4}$
M$_{\odot}$) seeds formed in the most massive ($> 10^{8}$ M$_{\odot}$) and rare
haloes; (ii) lighter ($\sim 10^{2}$ M$_{\odot}$) seeds to be found in less
massive and therefore more common haloes.
