We prove that if $k>1$ is an integer and $G$ is a finitely generated soluble group such that every infinite set of elements of $G$ contains a pair which generates a nilpotent subgroup of class at most $k$, then $G$ is an extension of a finite group by a torsion-free $k$-Engel group. As a corollary, there exists an integer $n$, depending only on $k$ and the derived length of $G$ , such that $G/Z_n\left(G\right)$ is finite. For $k<4$, such $n$ depends only on $k$.