We discuss the ground state properties of a Fermi gas with attractive interactions trapped in a two-dimensional optical lattice. By making use of the variational Monte Carlo method, we show that the system has an instability toward a density-wave state as well as a superfluid state. We also discuss how the additional staggered potential and inter-site Coulomb repulsion stabilize the supersolid region where a superfluid state coexists with a density-wave state. (C) 2009 Elsevier B.V. All rights reserved.