We investigate the size extensivity of the direct optimized effective potential procedure of Yang and Wu [Phys. Rev. Lett. 89, 143002 (2002)]. The choice of reference potential within the finite basis construction of the local Kohn-Sham potential can lead to a method that is not size extensive. Such a situation is encountered when one employs the Fermi-Amaldi potential, which is often used to enforce the correct asymptotic behavior of the exact exchange-correlation potential. The size extensivity error with the Fermi-Amaldi reference potential is shown to behave linearly with the number of electrons in the limit of an infinite number of well separated monomers. In practice, the error tends to be rather small and rapidly approaches the limiting linear behavior. Moreover, with a flexible enough potential basis set, the error can be decreased significantly. We also consider one possible reference potential, constructed from the van Leeuwen-Baerends potential, which provides a size extensive implementation while also enforcing the correct asymptotic behavior.