Context. The seeds of the first supermassive black holes may have resulted from the direct collapse of hot primordial gas in ≳10^{4 K} haloes, forming a supermassive or quasi-star 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 × 10^-2 AU and a mass of ~7 × 10^-2 M_⊙ 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 quasi-star of 10⁵ M_⊙ is expected to form within 10⁵ years.