We are concerned with the following Kirchhoff-type equation
$$ - \varepsilon ^2M\left( {\varepsilon ^{2 - N}\int_{{\open R}^N} {\vert \nabla u \vert^2{\rm d}x} } \right)\Delta u + V(x)u = f(u),\quad x \in {{\open R}^N},\quad N{\rm \ges }2,$$ where M ∈ C(ℝ+, ℝ+), V ∈ C(ℝN, ℝ+) and f(s) is of critical growth. In this paper, we construct a localized bound state solution concentrating at a local minimum of V as ε → 0 under certain conditions on f(s), M and V. In particular, the monotonicity of f(s)/s and the Ambrosetti–Rabinowitz condition are not required.