This paper investigates the use of relaxed recentered logarithmic barrier functions in the context of nonlinear model predictive control. These functions are a variation of the regular log-barrier functions that are introduced in the objective function of an optimization problem as a penalty for the deviation from the constraint set. The resulting MPC scheme has been studied in the case of linear dynamics. Several interesting results on the global nominal asymptotic stability of the corresponding closed-loop system and constraint satisfaction guarantees have been obtained. Extending them to the case of nonlinear dynamics is non-trivial, and we show in this paper that these properties can still hold locally. The theoretical results are demonstrated by the numerical implementation of a nonlinear benchmark system with four states and two inputs.