In this paper, we construct the solutions to the following nonlinear Schrödinger system -ϵ2Δu+P(x)u=μ1up+βup-12vp+12inRN,-ϵ2Δv+Q(x)v=μ2vp+βup+12vp-12inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} -\\epsilon ^{2}\\Delta u+P(x)u= \\mu _{1} u^{p}+\\beta u^{\\frac{p-1}{2}}v^{\\frac{p+1}{2}} \\ \\ \\ \ ext {in} \\ \\ \\mathbb {R}^{N},\\\\ -\\epsilon ^{2}\\Delta v+Q(x)v= \\mu _{2} v^{p}+\\beta u^{\\frac{p+1}{2}}v^{\\frac{p-1}{2}} \\ \\ \\ \ ext {in} \\ \\ \\mathbb {R}^{N}, \\end{array}\\right. } \\end{aligned}$$\\end{document}where 3< p<+infty , Nin {1,2}, epsilon >0 is a small parameter, the potentials P, Q satisfy 0<P_{0} le P(x)le P_{1} and Q(x) satisfies 0<Q_{0} le Q(x)le Q_{1}. We construct the solution for attractive and repulsive cases. When x_{0} is a local maximum point of the potentials P and Q and P(x_{0})=Q(x_{0}), we construct k spikes concentrating near the local maximum point x_{0}. When x_{0} is a local maximum point of P and overline{x}_{0} is a local maximum point of Q, we construct k spikes of u concentrating at the local maximum point x_{0} and m spikes of v concentrating at the local maximum point overline{x}_{0} when x_{0}ne overline{x}_{0}. This paper extends the main results established by Peng and Wang (Arch Ration Mech Anal 208:305–339, 2013) and Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016), where the authors considered the case N=3, p=3.