We establish the existence of positive solutions for a class of stationary Kirchhoff--type equations defined in the whole $\mathbb{R}^{3}$ involving critical growth in the sense of the Sobolev embedding and potentials which may decay to zero at infinity. We use minimax techniques combined with an appropriate truncated argument and a priori estimate. These results are new even for the local case which corresponds to nonlinear Schr\"{o}dinger equations.