Since very few type-I superconductors are known and most are elemental superconductors, there are very few experimental platforms where the interaction between type-I superconductivity and topologically nontrivial band structure can be probed. The rhenium aluminide Al₆Re has recently been identified as a type-I superconductor with a transition of 0.74 K and a critical field of ∼50 Oe. Here, we report its magnetotransport behavior including de Haas–van Alphen and Shubnikov–de Haas oscillations. Angular dependence of the magnetoresistance reveals a highly anisotropic Fermi surface with dominant hole character. From the strong oscillatory component ΔRxx in high magnetic fields up to 33 T, the Landau index infinite-field intercept in the case of a single oscillation frequency, and the phase factor φ where multiple frequencies coexist, are both ∼1/4. This intermediate value is suggestive of possible nontrivial band topology but does not allow strong conclusions.