  Let $t$ be a regular operator between Hilbert $C^*$-modules and $t^\dag$ be
its Moore-Penrose inverse. We investigate the Moore-Penrose invertibility of
the Gram operator $t^*t$. More precisely, we study some conditions ensuring
that $t^{\dag} = (t^* t)^{\dag} t^*= t^* (t t^*)^{\dag}$ and
$(t^*t)^{\dag}=t^{\dag}t^{* \dag}$ hold. As an application, we get some results
for densely defined closed operators on Hilbert $C^*$-modules over
$C^*$-algebras of compact operators.
