This work deals with online set-membership estimation problem of the state vector of linear discrete-time linear systems with sparse and delayed data. This problem is addressed in a bounded error context where both state disturbance and measurements noise are modeled by bounded boxes. A prediction-correction strategy is adopted to design a Luenberger-like observer where an outer approximation of the reachable set of the estimation error can be computed offline. In addition, by applying interval analysis and under a nonrestrictive observability assumption, both bounding and convergence features of the introduced interval-based state estimator are demonstrated. A numerical case study example is exposed to support the theoretical results by simulation results and to highlight the performance of the proposed estimation approach in the presence of sparse and delayed data.