As suspected by Yemon Choi the question is very closely related to the approximation property: As can be seen e.g. in the book Tensor Norms and Operator Ideals of Defant and Floret (page 64 combined with the remark 5.4) a Banach space $E$ has the approximation property if and only if the canonical mapping $E\tilde{\otimes}_\pi F \to (E' \otimes_\pi F')'$ is injective for all Banach spaces $F$ (or only for $F=E'$). I don't know if it is written somewhere but I would be very surprised if the mapping would always be injective for $F=E$.

The question as stated (whether $E\tilde{\otimes}_\pi E$ is isomorphic to some subspace of $E' \otimes_\pi E')'$ in a possibly non-canonical way) is, of course, very different.

As suspected by Yemon Choi the question is very closely related to the approximation property: As can be seen e.g. in the book Tensor Norms and Operator Ideals of Defant and Floret (page 64 combined with the remark 5.4) a Banach space $E$ has the approximation property if and only if the canonical mapping $E\tilde{\otimes}_\pi F \to (E' \otimes_\pi F')'$ is injective for all Banach spaces $F$ (or only for $F=E'$). I don't know if it is written somewhere but I would be very surprised if the mapping would always be injective for $F=E$.

The question as stated (whether $E\tilde{\otimes}_\pi E$ is isomorphic to some subspace of $(E' \otimes_\pi E')'$ in a possibly non-canonical way) is, of course, very different.