Context. Broad absorption lines (BALs) are present in the spectra of ~20% of quasars (QSOs); this indicates fast outflows (up to 0.2c) that intercept the observer's line of sight. These QSOs can be distinguished again into radio-loud (RL) BAL QSOs and radio-quiet (RQ) BAL QSOs. The first are very rare, even four times less common than RQ BAL QSOs. The reason for this is still unclear and leaves open questions about the nature of the BAL-producing outflows and their connection with the radio jet. Aims. We explored the spectroscopic characteristics of RL and RQ BAL QSOs with the aim to find a possible explanation for the rarity of RL BAL QSOs. Methods. We identified two samples of genuine BAL QSOs from SDSS optical spectra, one RL and one RQ, in a suitable redshift interval (2.5 <z <3.5) that allowed us to observe the Mg ii and Hβ emission lines in the adjacent near-infrared (NIR) band. We collected NIR spectra of the two samples using the Telescopio Nazionale Galileo (TNG, Canary Islands). By using relations known in the literature, we estimated the black-hole mass, the broad-line region radius, and the Eddington ratio of our objects and compared the two samples. Results. We found no statistically significant differences from comparing the distributions of the cited physical quantities. This indicates that they have similar geometries, accretion rates, and central black-hole masses, regardless of whether the radio-emitting jet is present or not. Conclusions. These results show that the central engine of BAL QSOs has the same physical properties with and without a radio jet. The reasons for the rarity of RL BAL QSOs must reside in different environmental or evolutionary variables.