To trace the origin of time-reversal symmetry breaking (TRSB) in Re-based superconductors, we performed comparative muon-spin rotation/relaxation (µSR) studies of superconducting noncentrosymmetric Re$$_{0.82}$$Nb$$_{0.18}$$ ($$T$$$$_c$$ = 8.8 K) and centrosymmetric Re ($$T$$$$_c$$ = 2.7 K). In Re$$_{0.82}$$Nb$$_{0.18}$$, the low-temperature superfluid density and the electronic specific heat evidence a fully-gapped superconducting state, whose enhanced gap magnitude and specific-heat discontinuity suggest a moderately strong electron-phonon coupling. In both Re$$_{0.82}$$Nb$$_{0.18}$$ and pure Re, the spontaneous magnetic fields revealed by zero-field µSR below $$T$$$$_c$$ indicate time-reversal symmetry breaking and thus unconventional superconductivity. The concomitant occurrence of TRSB in centrosymmetric Re and noncentrosymmetric Re$$T$$ ($$T$$ = transition metal), yet its preservation in the isostructural noncentrosymmetric superconductors Mg$$_{10}$$Ir$$_{19}$$B$$_{16}$$ and Nb$$_{0.5}$$Os$$_{0.5}$$, strongly suggests that the local electronic structure of Re is crucial for understanding the TRSB superconducting state in Re and Re$$T$$. We discuss the superconducting order parameter symmetries that are compatible with the observations.