To trace the origin of time-reversal symmetry breaking (TRSB) in Re-based superconductors, we performed comparative muon-spin rotation and relaxation (μSR) studies of superconducting noncentrosymmetric Re_0.82Nb_0.18 (T_c = 8.8 K) and centrosymmetric Re (T_c = 2.7 K). In Re_0.82Nb_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_0.82Nb_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 ReT (T = transition metal), yet its preservation in the isostructural noncentrosymmetric superconductors Mg₁₀Ir₁₉B₁₆ and Nb_0.5Os_0.5, strongly suggests that the local electronic structure of Re is crucial for understanding the TRSB superconducting state in Re and ReT. We discuss the superconducting order parameter symmetries that are compatible with the experimental observations.