The huge increase in the superconducting transition temperature of FeSe induced by an interface to SrTiO3 remains unexplained to date. However, there are numerous indications of the critical importance of specific features of the FeSe band topology in the vicinity of the Fermi surface. Here, we explore how the electronic structure of FeSe changes when located on another lattice matched substrate, namely a Si(001) surface, by first-principles calculations based on the density functional theory. We study non-magnetic (NM) and checkerboard anti-ferromagnetic (AFM) magnetic orders in FeSe and determine which interface arrangement is preferred. Our calculations reveal interesting effects of Si proximity on the FeSe band structure. Bands corresponding to hole pockets at the Γ point in NM FeSe are generally pushed down below the Fermi level, except for one band responsible for a small remaining hole pocket. Bands forming electron pockets centered at the M point of the Brillouin zone become less dispersive, and one of them is strongly hybridized with Si. We explain these changes by a redistribution of electrons between different Fe 3d orbitals rather than charge transfer to/from Si, and we also notice an associated loss of degeneracy between dxz and dyz orbitals.