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In the paper we study the existence of nonoscillatory solutions of the system $x_i^{\left(n\right)}\left(t\right)={\sum }_{j=1}^2{p}_{ij}\left(t\right)f_{ij}\left(x_j\left(h_{ij}\left(t\right)\right)\right),n\ge 2,i=1,2$, with the property $li{m}_{t\to \infty }{x}_i\left(t\right)/t^{k_i}=const\ne 0$ for some $k_i\in \{1,2,...,n-1\},i=1,2$. Sufficient conditions for the oscillation of solutions of the system are also proved.