We envisage the possibility of producing and tuning electronic orbital currents in atoms by weak, spatially inhomogeneous laser pulses with optical vortices. As already known, when interacting with such pulses an electronic system attains a definite amount of angular momentum. Here we explore this effect in the case when the system size is atomically small, i.e., much smaller than the scale of the inhomogeneity of the light pulses. We show that, nonetheless, angular momentum is transferred to the atomic system. The amount of the attained angular momentum depends on the position of the atom inside the field. Our numerical calculations and analytical analysis show that, indeed, light-induced orbital magnetism emerges in a dilute atomic gas when it is exposed to resonant or broadband optical vortices. This effect can be assessed, for instance, in a Faraday rotation experiment.