We report a comprehensive study of the centrosymmetric Re₃B and noncentrosymmetric Re₇B₃ superconductors. At a macroscopic level, their bulk superconductivity (SC), with T_c = 5.1K (Re₃B) and 3.3 K (Re₇B₃), was characterized via electrical-resistivity, magnetization, and heat-capacity measurements, while their microscopic superconducting properties were investigated by means of muon-spin rotation and relaxation (μSR). In both Re₃B and Re₇B₃ the low-T zero-field electronic specific heat and the superfluid density (determined via transverse-field μSR) suggest a nodeless SC. Both compounds exhibit some features of multigap SC, as evidenced by the temperature-dependent upper critical fields H_c2(T), as well as by electronic band-structure calculations. The absence of spontaneous magnetic fields below the onset of SC, as determined from zero-field μSR measurements, indicates a preserved time-reversal symmetry in the superconducting state of both Re₃B and Re₇B₃. Our results suggest that a lack of inversion symmetry and the accompanying antisymmetric spin-orbit coupling effects are not essential for the occurrence of multigap SC in these rhenium-boron compounds.