Using the virial exchange energy density defined by the integrand of the Levy-Perdew exchange virial relation, epsilon(x)(virial)[rho](r)=[3rho(r)+r.delrho(r)][deltaE(x)[rho] /deltarho(r)], it is shown that for spherical atoms with nuclear charge Z, the nuclear cusp of exchange energy density [(1/epsilon(x)(virial)) x (deltaepsilon(x)(virial)/deltar)](r-->0)= -(8Z/3). For the local density approximation, this condition is given by -10Z/3. Numerical results are presented for the rare gas atoms He-Xe using a variety of exchange-only local effective potentials. For the optimized effective potential and local density approximation the above conditions are obeyed quantitatively. It is found that the Krieger-Li-Iafrate approximation closely reproduces the optimized effective potential results, whereas those derived from the popular potentials due to Becke and Perdew-Wang give rise to much larger values. The exchange energy density defined analogously as the integrand of the directly calculated exchange energy of the model potential leads to the exact cusp values of -2Z for the optimized effective potential and -8Z/3 for the local density approximation.