We investigate the Maximally Abelian (MA) Projection for a single SU(2) instanton in continuum gauge theory. We find that there is a class of solutions to the differential MA gauge condition with circular monopole loops of radius R centered on the instanton of width ρ. However, the MA gauge fixing functional G decreases monotonically as R/ρ → 0. Its global minimum is the instanton in the singular gauge. We point out that interactions with nearby anti-instantons are likely to excite these monopole loops.