Let G be a group written multiplicatively.We say that G has the small sumsets property if for all positive integers r, s ≤ |G|, there exist subsets A, B ⊂ G such that |A| = r, |B| = s and |A · B| ≤ r + s − 1. If, in addition, it is possible to simultaneously satisfy A ⊂ B whenever r ≤ s, we speak of the nested small sumsets property for G. We prove that finite solvable groups satisfy this stronger form of the property. In the finite non-solvable case, we prove that subsets A, B ⊂ G satisfying |A| = r, |B| = s and |A· B| ≤ r +s−1 also exist, provided either r ≤ 12 or r + s ≥ |G| − 11.