Let $G$ be a simple graph with maximum degree $Δ(G)$ and chromatic index $χ'(G)$. A classic result of Vizing indicates that either $χ'(G )=Δ(G)$ or $χ'(G )=Δ(G)+1$. The graph $G$ is called $Δ$-critical if $G$ is connected, $χ'(G )=Δ(G)+1$ and for any $e\in E(G)$, $χ'(G-e)=Δ(G)$. Let $G$ be an $n$-vertex $Δ$-critical graph. Vizing conjectured that $α(G)$, the independence number of $G$, is at most $\frac{n}{2}$. The current best result on this conjecture, shown by Woodall, is that $α(G)<\frac{3n}{5}$. We show that for any given $\varepsilon\in (0,1)$, there exist positive constants $d_0(\varepsilon)$ and $D_0(\varepsilon)$ such that if $G$ is an $n$-vertex $Δ$-critical graph with minimum degree at least $d_0$ and maximum degree at least $D_0$, then $α(G)<(\frac{1}{2}+\varepsilon)n$. In particular, we show that if $G$ is an $n$-vertex $Δ$-critical graph with minimum degree at least $d$ and $Δ(G)\ge (d+2)^{5d+10}$, then \[ α(G) < \left. \begin{cases} \frac{7n}{12}, & \text{if $d= 3$; } \frac{4n}{7}, & \text{if $d= 4$; } \frac{d+2+\sqrt[3]{(d-1)d}}{2d+4+\sqrt[3]{(d-1)d}}n<\frac{4n}{7}, & \text{if $d\ge 19$. } \end{cases} \right. \]