  It has been recently claimed that primordial magnetic fields could relieve
the cosmological Hubble tension. We consider the impact of such fields on the
formation of the first cosmological objects, mini-halos forming stars, for
present-day field strengths in the range of $2\times 10^{-12}$ - $2\times
10^{-10}$ G. These values correspond to initial ratios of Alv\'en velocity to
the speed of sound of $v_a/c_s\approx 0.03 - 3$. We find that when $v_a/c_s\ll
1$, the effects are modest. However, when $v_a\sim c_s$, the starting time of
the gravitational collapse is delayed and the duration extended as much as by
$\Delta$z = 2.5 in redshift. When $v_a > c_s$, the collapse is completely
suppressed and the mini-halos continue to grow and are unlikely to collapse
until reaching the atomic cooling limit. Employing current observational limits
on primordial magnetic fields we conclude that inflationary produced primordial
magnetic fields could have a significant impact on first star formation,
whereas post-inflationary produced fields do not.
