  Possible variations of the quark-meson coupling (QMC) model are examined in
which the bag constant decreases in the nuclear medium. The reduction is
supposed to depend on either the mean scalar field or the effective mass of the
nucleon. It is shown that the electric and magnetic radii of the bound nucleon
are almost linearly correlated with the bag constant. Using the fact that the
size of the bound nucleon inside a nucleus is strongly constrained by
$y$-scaling data in quasielastic, electron-nucleus scattering, we set a limit
for the reduction allowed in the bag constant for these two models. The present
study implies that the bag constant can decrease up to 10--17 % at average
nuclear density, depending on the details of the model.
