Variational methods to treat a many-electron system embedded in the environment, which is represented by means of only its electron density, are considered. It is shown that the embedding operator is a local potential in the case where the electron-electron repulsion is treated exactly and the trial embedded wave function takes the multideterminantal form with a fixed number of determinants. The local embedding potential is constructed by imposing that it leads to the same electron density as the one which minimizes the Hohenberg-Kohn functional. For the limiting cases of single-determinant and configuration interaction forms of the embedded wave function, the expressions for the local embedding potential using commonly known density functionals are given. The relation between the derived local embedding potential and the effective embedding potential in the case of the embedded Kohn-Sham system [T. A. Wesolowski and A. Warshel, J. Phys. Chem. 97, 8050 (1993)] is discussed in detail.