  Supermassive black holes are not only common in the present-day galaxies, but
billion solar masses black holes also powered $z\geq 6$ quasars. One efficient
way to form such black holes is the collapse of a massive primordial gas cloud
into a so-called direct collapse black hole. The main requirement for this
scenario is the presence of large accretion rates of $\rm \geq
0.1~M_{\odot}/yr$ to form a supermassive star. It is not yet clear how and
under what conditions such accretion rates can be obtained. The prime aim of
this work is to determine the mass accretion rates under non-isothermal
collapse conditions. We perform high resolution cosmological simulations for
three primordial halos of a few times $\rm 10^7~M_{\odot}$ illuminated by an
external UV flux, $\rm J_{21}=100-1000$. We find that a rotationally supported
structure of about parsec size is assembled, with an aspect ratio between $\rm
0.25 - 1$ depending upon the thermodynamical properties. Rotational support,
however, does not halt collapse, and mass inflow rates of $\rm \sim
0.1~M_{\odot}/yr$ can be obtained in the presence of even a moderate UV
background flux of strength $\rm J_{21} \geq 100$. To assess whether such large
accretion rates can be maintained over longer time scales, we employed sink
particles, confirming the persistence of accretion rates of $\rm \sim
0.1~M_{\odot}/yr$. We propose that complete isothermal collapse and molecular
hydrogen suppression may not always be necessary to form supermassive stars,
precursors of black hole seeds. Sufficiently high inflow rates can be obtained
for UV flux $\rm J_{21}=500-1000$, at least for some cases. This value brings
the estimate of the abundance of direct collapse black hole seeds closer to
that high redshift quasars.
