We study, both numerically and analytically, the effect of a substrate on the optical properties of high-refractive-index dielectric nanoparticles. We demonstrate that the optical response of subwavelength nanoparticles can be effectively modeled in the dipole approximation when the dielectric particle is replaced by a pair of point electric and magnetic dipoles with modified polarizabilities. Based on our model, we reveal the existence of the substrate-induced bianisotropic interaction (magnetoelectric coupling) between electric and magnetic dipole modes, which is strongly enhanced near the corresponding dipole resonances. This results in a specific feature of the radiation pattern that can be employed to identify the effective bianisotropic interaction experimentally.