In this paper, we extend the idea of a tube-based economic MPC framework for linear systems by taking linear average constraints into account. Using a specifically defined integral stage cost, we can explicitly consider the influence of the disturbance. The satisfaction of the average constraints is guaranteed despite the disturbances acting on the system, and we show results on optimal steady-state operation. Moreover, we make use of two algorithmic approaches from tube-based MPC which provide differences in the closed-loop behavior and in the average performance.