Context. The potential importance of magnetic fields during structure formation and gravitational collapse in the early Universe has been shown in several studies. In particular, magnetic field amplification by the small-scale dynamo plays an important role in addition to the pure amplification expected from gravitational collapse. Aims. In this paper we study the small-scale dynamo for halos of greater than or similar to 10(7) M-circle dot collapsing at z greater than or similar to 12, under different ambient conditions due to the strength of the Lyman-Werner background. Additionally, we estimate the approximate saturation level by varying the initial magnetic field strength. Methods. We performed cosmological magnetohydrodynamical simulations for three distinct halos of similar to 10(7) M-circle dot at z >= 12 by varying the Jeans resolution from 32 - 256 cells and employed Lyman Werner background flux of strengths 10(2) - 10(5) in units of J(21), where J(21) = 10(-21) erg cm(-2) sr(-1) s(-1)Hz(-1). To follow the chemical and thermal evolution of the gas, we made use of the KROME package. Results. In addition to the compression by collapse, we find magnetic field amplification via the dynamo in the regimes of atomic and molecular hydrogen cooling. Moreover, we find a lower saturation level in the molecular hydrogen cooling regime. This behaviour can be understood in terms of the generally reduced radial infall velocities and vorticities in this regime, as well as the higher Mach numbers of the gas, which give rise to a smaller saturation ratio. Conclusions. Our results overall suggest that the dynamo operates over a large range of conditions in the collapsing gas.