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.

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