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In this paper, we study the following nonlinear Schrödinger–Newton type system:\left\{\begin{aligned} &\displaystyle{-}\epsilon^{2}\Delta u+u-\Phi(x)u=Q(x)|u% |u,&&\displaystyle x\in\mathbb{R}^{3},\\ &\displaystyle{-}\epsilon^{2}\Delta\Phi=u^{2},&&\displaystyle x\in\mathbb{R}^{% 3},\end{aligned}\right.where {\epsilon>0} and {Q(x)} is a positive bounded continuous potential on {\mathbb{R}^{3}} satisfying some suitable conditions. By applying the finite-dimensional reduction method, we prove that for any positive integer k, the system has a positive solution with k-peaks concentrating near a strict local minimum point {x_{0}} of {Q(x)} in {\mathbb{R}^{3}}, provided that {\epsilon>0} is sufficiently small.

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