In this paper, we study the following Kirchhoff-type equation:

$$M\left({\displaystyle {\int }_{{\mathbb{R}}^2}(|\mathrm{\nabla }u{|}^2+u^2)\mathrm{d}x}\right)(-\mathrm{\Delta }u+u)+\mu V(x)u=K(x)f(u)\mathrm{i}\mathrm{n}{\mathbb{R}}^2,$$

where $M\in C({\mathbb{R}}^{+},{\mathbb{R}}^{+})$ is a general function, $V\ge 0$ and its zero set may have several disjoint connected components, $\mu >0$ is a parameter, $K$ is permitted to be unbounded above, and $f$ has exponential critical growth. By using the truncation technique and developing some approaches to deal with Kirchhoff-type equations with critical growth in the whole space, we get the existence and concentration behavior of solutions. The results are new even for the case $M\equiv 1$.