We study the $G_2$ analogue of the Goldberg conjecture on non-compact solvmanifolds. In contrast to the almost-K��hler case we prove that a 7-dimensional solvmanifold cannot admit any left-invariant calibrated $G_2$-structure $��$ such that the induced metric $g_��$ is Einstein, unless $g_��$ is flat. We give an example of 7-dimensional solvmanifold admitting a left-invariant calibrated $G_2$-structure $��$ such that $g_��$ is Ricci-soliton. Moreover, we show that a 7-dimensional (non-flat) Einstein solvmanifold $(S,g)$ cannot admit any left-invariant cocalibrated $G_2$-structure $��$ such that the induced metric $g_�� = g$.

21 pages. To appear in The Asian Journal of Mathematics