Let $X$ be a Banach space with the Grothendieck property, $Y$ a reflexive Banach space, and let $X\check{\otimes}_{\varepsilon} Y$ be the injective tensor product of $X$ and $Y$. \item {(a)} If either $X^{\ast \ast }$ or $Y$ has the approximation property and each continuous linear operator from $X^\ast $ to $Y$ is compact, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property. \item {(b)} In addition, if $Y$ has an unconditional finite dimensional decomposition, then $X\check{\otimes}_{\varepsilon} Y$ has the Grothendieck property if and only if each continuous linear operator from $X^\ast $ to $Y$ is compact.