The answer is no. $\mathcal Hom (\mathcal O_X,\omega)\simeq \omega$, but $\omega\otimes \mathcal Hom (\omega, \mathcal O_X)$ is not necessarily reflexive, let alone locally free. However, it is true that if $X$ is $G_1$, that is, Gorenstein in codimension $1$, then $$ (\omega\otimes \mathcal Hom (\omega, \mathcal O_X))^{**}\simeq \mathcal O_X. $$
EDIT: (due to popular demand, here is a better formed statement for the CM is not even needed part of the original)
This is actually true in a little more general setting: For simplicity assume that $X$ is $G_1$, $S_2$, equidimensional of dimension $d$ and admits a dualizing complex denoted by $\omega_X^\cdot$. (If, say, by variety you mean a quasi-projective (reduced) scheme of finite type over a field, then the latter assumption is automatic. If in addition you also mean irreducible, then so is the former.)
Let $$\omega_X := h^{-d}(\omega_X^\cdot)$$ and $$\omega_X^*:=\mathcal Hom (\omega_X, \mathcal O_X).$$
Then $${(\omega_X\otimes \omega_X^*)}^{**}\simeq \mathcal O_X$$