So here is a sort of vague question. Suppose $\sum_{i=1}^kx_i\otimes y_i\in X\otimes Y$ be a general element in the tensor product of finite dimensional vector spaces $X$ and $Y$. Does there exist finite dimensional vector spaces $\tilde{X}$ and $\tilde{Y}$ such that $X\otimes Y$ sits inside $\tilde{X}\otimes \tilde{Y}$ and that $\sum_{i=1}^kx_i\otimes y_i$ is $\tilde{x}\otimes \tilde{y}$ for some $\tilde{x}\in \tilde{X}$ and for some $\tilde{y}\in\tilde{Y}$.
I would like some hints rather than a complete answer.
Well, you could always take $\tilde{X}=X\otimes Y$...