The compound nuclei ⁵⁸Co* and ⁶¹Ni* have been populated at overlapping excitation energies by transfer reactions ⁵⁶Fe(⁶Li, α) ⁵⁸Co* (surrogate of n + ⁵⁷Co) at E_lab = 35.9 MeV and ⁵⁹Co(⁶Li, α) ⁶¹Ni* (surrogate of n + ⁶⁰Ni) at E_lab = 40.5 MeV, respectively. The ⁵⁷Co(n, xp) cross sections in the equivalent neutron energy range of 8.6–18.8 MeV have been determined within the framework of surrogate reaction ratio method using ⁶⁰Ni(n, xp) cross section values from the literature as reference. The proton decay probabilities of the compound systems have been determined by measuring evaporated protons at backward angles in coincidence with projectile-like fragments detected around the grazing angle. The measured ⁵⁷Co(n, xp) cross sections are in good agreement with both the predictions of TALYS-1.8 statistical model code with default parameters using different microscopic level densities and data evaluation library JEFF-3.3 up to equivalent neutron energy ≈12.6 MeV, while for higher energies the measured ⁵⁷Co(n, xp) cross sections are found to be consistently higher than the predictions. However, the TALYS-1.8 calculations with modified values of input potential parameters provide a reasonable reproduction of the measured ⁵⁷Co(n, xp) cross sections for the entire neutron energy range. The observed discrepancies at higher energies between the experimental data and the predictions of both the JEFF-3.3 library and the TALYS-1.8 calculations with default parameters indicate the need of new evaluations for this reaction.