Let $T=U|T|$ and $S=V|S|$ be the polar decompositions of adjointable operators $T$ and $S$, respectively on a Hilbert $C^*$-module. We determine these pairs of operators for which their products $TS$ accepts the polar decomposition as $TS=UV|TS|$. Specially, we provide sufficient conditions for a certain operator $T$ such that its Aluthge transform $\tilde T=|T|^{1/2}U|T|^{1/2}$ admits the polar decomposition.