Zero-field and transverse-field muon spin rotation/relaxation (μSR) experiments were undertaken in order to elucidate the microscopic properties of a strongly coupled superconductor Mo₈Ga₄₁ with T_c = 9.8 K. The upper critical field extracted from the transverse-field μSR data exhibits significant reduction with respect to the data from thermodynamic measurements indicating the coexistence of two independent length scales in the superconducting state. Accordingly, the temperature-dependent magnetic penetration depth of Mo₈Ga₄₁ is described using a model in which two s wave superconducting gaps are assumed. A V for Mo substitution in the parent compound leads to the complete suppression of one superconducting gap, and Mo₇VGa₄₁ is well described within the single s wave gap scenario. The reduction in the superfluid density and the evolution of the low-temperature resistivity upon V substitution indicate the emergence of a competing state in Mo₇VGa₄₁ that may be responsible for the closure of one of the superconducting gaps.