Bounded Engel elements in groups satisfying an identity

We prove that a residually finite group G satisfying an identity w ≡ 1 and generated by a commutator closed set X of bounded left Engel elements is locally nilpotent. We also extend such a result to locally graded groups, provided that X is a normal set. As an immediate consequence, we obtain that a locally graded group satisfying an identity, all of whose elements are bounded left Engel, is locally nilpotent