  Context. The seeds of the first supermassive black holes may have resulted
from the direct collapse of hot primordial gas in $\gtrsim 10^4$ K haloes,
forming a supermassive or quasistar as an intermediate stage.
  Aims. We explore the formation of a protostar resulting from the collapse of
primordial gas in the presence of a strong Lyman-Werner radiation background.
Particularly, we investigate the impact of turbulence and rotation on the
fragmentation behaviour of the gas cloud. We accomplish this goal by varying
the initial turbulent and rotational velocities.
  Methods. We performed 3D adaptive mesh refinement simulations with a
resolution of 64 cells per Jeans length using the ENZO code, simulating the
formation of a protostar up to unprecedentedly high central densities of
$10^{21}$ cm$^{-3}$, and spatial scales of a few solar radii. To achieve this
goal, we employed the KROME package to improve modelling of the chemical and
thermal processes.
  Results. We find that the physical properties of the simulated gas clouds
become similar on small scales, irrespective of the initial amount of
turbulence and rotation. After the highest level of refinement was reached, the
simulations have been evolved for an additional ~5 freefall times. A single
bound clump with a radius of $2 \times 10^{-2}$ AU and a mass of ~$7 \times
10^{-2}$ M$_{\odot}$ is formed at the end of each simulation, marking the onset
of protostar formation. No strong fragmentation is observed by the end of the
simulations, regardless of the initial amount of turbulence or rotation, and
high accretion rates of a few solar masses per year are found.
  Conclusions. Given such high accretion rates, a quasistar of $10^5$
M$_{\odot}$ is expected to form within $10^5$ years.
