We have studied theoretically the interaction of an isolated Cu atom adsorbed on the oxygen sites of the regular MgO (001) surface with the aim of providing an accurate estimate of the adhesion energy. We performed cluster model calculations using a variety of first principles quantum-chemical approaches; local (spin) density approximation [L(S)DA], density functionals that include density gradient corrections (GC-DF), hybrid density functional (B3LYP), and explicitly correlated wave functions. Various combinations of exchange-correlation functionals and different methods to introduce electron correlation, including MP2 and CCSD(T), have been considered. The dependence of the results on cluster and basis set size has been carefully checked. We found that the hybrid DF method, B3LYP, and explicitly correlated wave functions, CCSD(T), give similar results with an adhesion energy of about 0.40+/-0.05 eV; GC-DF methods suggest a higher binding energy of 0.6 eV. Therefore, Cu atoms can be considered to bind to oxygen centers of the ideal MgO (001) surface in a relatively weak chemical fashion, in broad agreement with the low sticking coefficient measured experimentally.