We present a first-principles study of electronic and magnetic properties of thin Co films on a BaTiO₃(0 0 1) single crystal. The crystalline structure of 1-3 monolayer thick Co films was determined and served as input for calculations of the electronic and magnetic properties of the films. The estimation of exchange constants indicates that the Co films are ferromagnetic with a high critical temperature, which depends on the film thickness and the interface geometry. In addition, we calculated x-ray absorption spectra, related magnetic circular dichroism (XMCD) and linear dichroism (XLD) of the Co L_{2, 3} edges as a function of Co film thickness and ferroelectric polarization of BaTiO₃. We found characteristic features, which depend strongly on the magnetic properties and the structure of the film. While there is only a weak dependence of XMCD spectra on the ferroelectric polarization, the XLD of the films is much more sensitive to the polarization switching, which could possibly be observed experimentally.