A nonlinear thermodynamic theory is developed for the strain-mediated direct magnetoelectric (ME) effect displayed by ferroelectric-ferromagnetic nanostructures. This effect results from transmission of magnetic-field-induced deformations of a thick ferromagnetic substrate to a thin ferroelectric overlayer, where the polarization changes due to lattice strains. The strain-dependent polarization and permittivity of an epitaxial nanolayer (few tens of nm thick) are calculated using the thermodynamic theory of single-domain ferroelectric films. The substrate magnetostrictive deformations are described phenomenologically, taking into account their nonlinear variation with magnetic field. The calculations show that ME polarization and voltage coefficients strongly depend on the initial strain state of the film. For BaTiO(3) and PbTiO(3) films deposited on Co(0.8)Zn(0.2)Fe(2)O(4), the out-of-plane polarization and related ME coefficients are calculated numerically as a function of magnetic field parallel to the interface. For films stabilized in the monoclinic phase, this transverse ME response depends on the orientation of magnetic field relative to their in-plane crystallographic axes. The longitudinal ME coefficient is also evaluated and, for a substrate geometry minimizing the demagnetizing field, predicted to be comparable to the transverse one. For BaTiO(3) and PbTiO(3) films deposited on Terfenol-D, the calculations yield high ME polarization coefficients approximately 10(-7) s m(-1) and giant ME voltage coefficients approximately 50 V cm(-1) Oe(-1).