I was sure it is known, but it appears to be an open problem (see the answer of Terry Tao).

Assume for a group $G$ there is a polynomial $P$ such that given $n\in\mathbb N$ there is set of generators $S=S^{-1}$ such that $$|S^n|\leqslant P(n)\\cdot|S|\ \ (\text{or stronger condition}\ |S^k|\leqslant P(k)\\cdot|S|\ \text{for all}\ k\le n) .$$ Then $G$ is virtually nilpotent (or equivalently it has polynomial growth).

**Comments:**

Note that typically, $|S|\to \infty$ as $n \to\infty$ (otherwise it follows easily from the original Gromov's theorem).

If it is known, then it would give a group-theoretical proof that manifolds with almost non-negative Ricci curvature have virtually nilpotent fundamental group (see Kapovitch--Wilking, "Structure of fundamental groups..."). This proof would use only one result in diff-geometry: Bishop--Gromov inequality.