Tensor power of a quotient space

Question

Let $E$ and $F$ be vector spaces, $F$ a subspace of $E$. Is there any canonical isomorphism between $E/F \otimes E/F$ and a quotient of the form $E \otimes E/G$, where $G$ is a subspace of $E \otimes E$ depending only on $F$ and $E$ ?

Also, what if we use symmetric tensor products ?