A localized version of the Method of Fundamental Solutions is applied to the 2D steady heat transfer equation with spatially varying thermal conductivity. Though the corresponding fundamental solution cannot be computed in general, the localization splits the original problem into several subproblems defined on small subdomains; in these subdomains, the subproblems are approximately converted to certain convection-diffusion equations with constant coefficients, so that the Method of Fundamental Solutions is applicable. In each subdomain, the corresponding subproblem is solved separately. This results in an iterative method, which mimics the overlapping (alternating) Schwarz method. Due to its advantageous numerical properties, the technique seems a useful generalization of the Method of Fundamental Solutions. The method is illustrated through numerical examples.