The purpose of this paper is to introduce and study twisted O-operators on
3-Lie algebras. We construct an L?-algebra whose Maurer-Cartan elements are
twisted O-operators and define a cohomology of a twisted O-operator T as the
Chevalley-Eilenberg cohomology of a certain 3-Lie algebra induced by T with
coefficients in a suitable representation. Then we consider infinitesimal
and formal deformations of twisted O-operators.