The refined values of the magnetic dipole moments of Re185 and Re187 nuclei are obtained. For this, we perform a combined relativistic coupled cluster and density-functional theory calculation of the shielding constant for the ReO4- anion. In this calculation, we explicitly include the effect of the finite nuclear magnetization distribution in the single-particle nuclear model using the Woods-Saxon potential for the valence nucleon. By combining the obtained value of the shielding constant σ=4069(389) ppm with the available experimental nuclear magnetic resonance data we obtain the values: μ(Re185)=3.1567(3)(12)μN, μ(Re187)=3.1891(3)(12)μN, where the first uncertainty is the experimental one and the second is due to theory. The refined values of magnetic moments are in disagreement with the tabulated values, μ(Re185)=3.1871(3)μN, μ(Re187)=3.2197(3)μN, which were obtained using the shielding constant value calculated for the atomic cation Re7+ rather than the molecular anion. The updated values of the nuclear magnetic moments resolve the disagreement between theoretical predictions of the hyperfine structure of H-like rhenium ions which were based on the tabulated magnetic moment values and available experimental measurements. Using these experimental data we also extract the value of the parameter of nuclear magnetization distribution introduced in [J. Chem. Phys. 153, 114114 (2020)0021-960610.1063/5.0024103], which is required to predict hyperfine structure constants for rhenium compounds.