Bilevel optimization problems embed the optimality conditions of a sub-problem into the constraints of another optimization problem. We introduce the concept of near-optimality robustness for bilevel problems, protecting the upper-level solution feasibility from limited deviations at the lower level. General properties and necessary conditions for the existence of solutions are derived for near-optimal robust versions of generic bilevel problems. A duality-based solution method is defined when the lower level is convex, leveraging the methodology from the robust and bilevel literature. Numerical results assess the efficiency of the proposed algorithm and the impact of valid inequalities on the solution time.