The existence of at least one solution for a nonlocal system \(\begin{aligned} \begin{aligned} -a_{i}\left( \int _ \Omega |u_{i}| ^{q_i}\right) \Delta u_{i} =h_i (x, u_{1}, \ldots , u_{n}) f_i \left(\int _ \Omega |u_{i}| ^{p_i}\right)\\+\delta _i \left(x, u_{1}, \ldots , u_{n}\right) g_i\left(\int _ \Omega |u_{i}| ^{r_i}\right), \end{aligned} \end{aligned}\)on a bounded domain of \({\mathbb {R}}^N, N\ge 1\), under Dirichlet boundary conditions, is proved, where \( p_i, q_i, r_i \in [1, +\infty )\) are some given constants, \(h_i, \delta _i: {\overline{\Omega }} \times \prod _{i=1}^{n} {\mathbb {R}} \rightarrow {\mathbb {R}}^{+}\) are Carathéodory functions and \(a_{i}, f_i, g_i: [0, +\infty ) \longrightarrow (0, +\infty )\) are continuous functions, for \(i=1,\ldots , n\). The method is based on a combination of sub- and supersolution method with the pseudomonotone operators theory. In addition, the existence of solutions to a system of singular nonlocal problems is done.