We consider gradient estimates for $H^1$ solutions of linear elliptic systems in divergence form $\partial _{\alpha }(A_{ij}^{\alpha \beta } \partial _{\beta } u^{j}) = 0$. It is known that the Dini continuity of coefficient matrix $A = (A_{ij}^{\alpha \beta })$ is essential for the differentiability of solutions. We prove the following results:

(a) If A satisfies a condition slightly weaker than Dini continuity but stronger than belonging to VMO, namely that the $L^2$ mean oscillation $\omega_{A,a}$ of A satisfies

$$ X_{A,2} := \limsup\limits_{r\rightarrow 0} r {{\int \limits }_{r}^{2}} \frac {\omega _{A,2}(t)}{t^{2}} \exp \left (C_{*} {{\int \limits }_{t}^{R}} \frac {\omega _{A,2}(s)}{s} ds\right ) dt < \infty , $$

where $C_*$ is a positive constant depending only on the dimensions and the ellipticity, then ∇u ∈ BMO.

(b) If $X_{A,2}$ = 0, then ∇u ∈ V MO.

(c) Finally, examples satisfying $X_{A,2}$ = 0 are given showing that it is not possible to prove the boundedness of ∇u in statement (b), nor the continuity of ∇u when $\nabla u \in L^{\infty } \cap VMO$.