  We report a comprehensive study of the noncentrosymmetric NbReSi
superconductor by means of muon-spin rotation and relaxation ($\mu$SR) and
nuclear magnetic resonance (NMR) techniques. NbReSi is a bulk superconductor
with $T_c = 6.5$ K, characterized by a large upper critical field, which
exceeds the Pauli limit. Both the superfluid density $\rho_\mathrm{sc}(T)$
(determined via transverse-field $\mu$SR) and the spin-lattice relaxation rate
$T_1^{-1}(T)$ (determined via NMR) suggest a nodeless superconductivity (SC) in
NbReSi. We also find signatures of multigap SC, here evidenced by the
field-dependent muon-spin relaxation rate and the electronic specific-heat
coefficient. The absence of spontaneous magnetic fields below $T_c$, as evinced
from zero-field $\mu$SR measurements, indicates a preserved time-reversal
symmetry in the superconducting state of NbReSi. Finally, we discuss possible
reasons for the unusually large upper critical field of NbReSi, most likely
arising from its anisotropic crystal structure.
