  Let $g$ be an element of a group $G$. For a positive integer $n$, let
$E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots ,g]$
over $x\in G$, where $g$ is repeated $n$ times. We prove that if $G$ is a
profinite group such that for every $g\in G$ there is $n=n(g)$ such that
$E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$
is locally nilpotent. The proof uses the Wilson--Zelmanov theorem saying that
Engel profinite groups are locally nilpotent. In the case of a finite group
$G$, we prove that if, for some $n$, $|E_n(g)|\leq m$ for all $g\in G$, then
the order of the nilpotent residual $\gamma _{\infty}(G)$ is bounded in terms
of $m$.
