This work addresses the problem of reachable set-based consensus for positive multiagent systems affected by typical classes of bounded disturbances. In the presence of disturbances, a reachable set-based consensus with positivity preservation is proposed to ensure that the state of the closed-loop system remains positive while enclosing the reachable set of the defined consensus error with an ellipsoidal bounding region. Sufficient conditions are established to achieve the reachable set-based positive consensus under the energy-bounded or peak-bounded disturbance input. Equivalent design conditions are provided, for which a heuristic algorithm is proposed for computing and optimizing the bounding region. Simulations are conducted to validate the obtained results.