Despite its unique structural features, the magnetism of single-layered cuprate with five oxygen coordination (T *-type structure) has not been investigated thus far. Here, we report the results of muon spin relaxation and magnetic susceptibility measurements to elucidate the magnetism of T *-type La 1−x/2 Eu 1−x/2 SrxCuO4 (LESCO) via magnetic Fe-and non-magnetic Zn-substitution. We clarified the inducement of the spin-glass (SG)-like magnetically ordered state in La 1−x/2 Eu 1−x/2 SrxCuyFe1−yO4 with x = 0.24 + y, and the non-magnetic state in La 1−x/2 Eu 1−x/2 SrxCuyZn1−yO4 with x = 0.24 after the suppression of superconductivity for y ≥ 0.025. The SG state lies below ∼7 K in a wide Sr concentration range between 0.19 and 0.34 in 5% Fe-substituted LESCO. The short-range SG state is consistent with that originating from the Ruderman-Kittel-Kasuya-Yosida interaction in a metallic state. Thus, the results provide the first evidence for Fermi liquid (FL) state in the pristine T *-type LESCO. Taking into account the results of an oxygen K-edge X-ray absorption spectroscopy measurement [J. Phys. Soc. Jpn. 89, 075002 (2020)] reporting the actual hole concentrations in LESCO, our results demonstrate the existence of the FL state in a lower hole-concentration region, compared to that in T -type La2−xSrxCuO4. The emergence of the FL state in a lower hole-concentration region is possibly associated with a smaller charge transfer gap energy in the parent material with five oxygen coordination.