  We give a construction of Gorenstein projective $\tau$-tilting modules in
terms of tensor products of modules. As a consequence, we give a class of
non-self-injective algebras admitting non-trivial Gorenstein projective
$\tau$-tilting modules. Moreover, we show that a finite dimensional algebra
$\Lambda$ over an algebraically closed field is $CM$-$\tau$-tilting finite if
$T_n(\Lambda)$ is $CM$-$\tau$-tilting finite which gives a partial answer to a
question on $CM$-$\tau$-tilting finite algebras posed by Xie and Zhang.
