Minimization of the Hohenberg-Kohn total energy functional EHK [�] in the presence of the constraint � - �_B  ≥ 0, where �_B is some arbitrarily chosen electron density comprising integer number of electrons is considered. To access better numerical accuracy of approximations to EHK [�] in practice, the search for optimal � - �_B is performed using auxiliary quantities such as orbitals of a reference system of non-interacting electrons [Wesolowski and Warshel, J Phys Chem 1993, 97, 8050] or a wavefunction-like object corresponding to interacting electrons [Wesolowski, Phys Rev A 2008, 77, 012504]. In both cases, the condition � - �_B  ≥ 0 leads to a local potential (orbital-free effective embedding potential) of the same general form if expressed by means of universal density functionals. In this work, it is shown that the same local potential is obtained if the search for optimal � - �_B is performed among one-particle reduced density matrices.