The correlation functional Ec[ρ] known in Levy's constrained search formulation of density functional theory is also one of the components of the energy functional in frozen-density embedding theory (FDET) [Wesolowski Phys. Rev. A, 77, 2008, 012504] if the embedded wave function has the single-determinant form. The relation between the FDET energy and quantities available from an auxiliary system is derived. In the auxiliary system, Ec[ρ] and its functional derivative (correlation potential) are entirely neglected. The relation is exact up to the quadratic term in density changes caused by electron-electron correlation. It is discussed in view of its practical applications in modeling an electronic structure of embedded species.