In this paper, we focus on the following nonlinear Schr\”odinger equations with linear couples \begin{equation*} \begin{cases} -\Delta u+V_1(x)u+\lambda_1u=\mu _1\int\limits_{\mathbb{R}^{3}}\frac{|u(y)|^p}{|x-y|}\,dy|u|^{p-2}u+\beta v \hspace{5mm} \text{in} \hspace{1mm}\mathbb R^{3},\\ -\Delta v+V_2(x)v+\lambda_2v=\mu _2\int\limits_{\mathbb{R}^{3}}\frac{|v(y)|^q}{|x-y|}\,dy|v|^{q-2}v+\beta u \hspace{5mm} \text{in} \hspace{1mm}\mathbb R^{3},\\ \int\limits_{\mathbb{R}^{3}}|u|^2\,dx=a, \hspace{1mm} \int\limits_{\mathbb{R}^{3}}|v|^2\,dx=b, \end{cases} \end{equation*} where $\frac{5}{3}0$, $a,b\geq 0$, $\beta \in \mathbb{R}\setminus\{0\}$, $\lambda_1,\lambda_2\in\mathbb{R}$ are Lagrange multipliers and $V_1(x),V_2(x):\mathbb{R}^{3}\to \mathbb{R}$ are trapping potentials. We prove the existence of the solutions with prescribed $L^2(\mathbb{R})$-norm with trivial trapping potentials and nontrivial trapping potentials by applying the rearrangement inequalities.