The k-dimensional system of neutral type nonlinear difference equations with delays in the following form
\begin{equation*}
\begin{cases}
\Delta \Big(x_i(n)+p_i(n)\,x_i(n-\tau_i)\Big)=a_i(n)\,f_i(x_{i+1}(n-\sigma_i))+g_i(n),\\
\Delta \Big(x_k(n)+p_k(n)\,x_k(n-\tau_k)\Big)=a_k(n)\,f_k(x_1(n-\sigma_k))+g_k(n),
\end{cases}
\end{equation*}
where i=1,…,k−1, is considered. The aim of this paper is to present sufficient conditions for the existence of nonoscillatory bounded solutions of the above system with various (pi​(n)), i=1,…,k, k≥2