The coexistence of magnetism and ferroelectricity in the same crystalline phase of a so-called multiferroic (MF) material involves the opportunity of magnetoelectric (ME) coupling. ME coupling, in principle, offers magnetization switching by an electric field or polarization switching by a magnetic field. Since this phenomenon allows to store information in nanometer-sized memories with four logic states, the issues of MFs are of prime interest. In the single-phase MFs, however, the electric polarization and magnetization interact weakly with each other while ferromagnetism disappears far below room temperature. A more robust scenario of magnetoelectricity might occur in artificial MFs composed of ferromagentic (FM) thin films which are grown epitaxially on a ferroelectric substate. In the study of composite MFs, the reults of ab initio calculations have shown an extremely promising direction for the next years. Although these calculations go ahead of experiment they explore the trends and basic physics of ME. Here, on the basis of first-principles calculations we predict that epitaxial ultrathin Fe films deposited on TiO₂-terminated (001) surface of ATiO₃ perovskites (A = Pb, Ba) exhibit an unexpected change in their magnetic structure with increasing Fe-film thickness. The magnetic order changes from strongly FM for the single-monolayer Fe system to ferrimagnetic with almost vanishing magnetization upon deposition of a second Fe layer. FM order is restored for thicker Fe films. This effect can be understood in terms of hybridization of electronic states and structural relaxation. Additionally, we study the effect of iron oxidation on the ME coupling at the Fe₂/ATiO₃(001) interface. The oxygen coverage ranged between 0.5 and 2.0 adsorbed O atom per Fe atom. The magnetic properties of the Fe layer are gradually degraded with increasing O coverage. However, the change in magnetization which is induced by the electric polarization reversal remains robust for all energetically favorable compositions. Thus, the surface oxidation of composite MFs cannot destroy the switchable magnetoelectricity.