Lean NOx trap (LNT) is one of the most effective after-treatment technologies used to reduce NOx emissions of diesel engines. One relevant problem in this context is LNT regeneration timing control. This problem is indeed difficult due to the fact that LNTs are highly nonlinear systems, involving complex physical/chemical processes, that are hard to model. In this paper, a novel approach for regeneration timing of LNTs is proposed, allowing us to overcome these issues. This approach, named data-driven model predictive control (D2-MPC), does not require a physical model of the engine/trap system but is based on low-complexity polynomial prediction models, directly identified from data. The regeneration timing is computed through an optimization algorithm, which uses the identified models to predict the LNT behavior. Two D2-MPC strategies are proposed, and tested in a co-simulation study, where the plant is represented by a detailed LNT model, built using the well-known commercial tool AMEsim, and the controller is implemented in matlab/simulink.