This technical note studies the consensus problems under saturation constraints in interconnection states, both the componentwise and the norm bounded constraints. By utilizing the edge Laplacian and the state saturation, the networked linear systems under interconnection constraints are described by the node and constrained edge dynamics. Sufficient conditions are identified under which the given constrained systems achieve the average consensus. Then, as an example of applications, we extend the analytical result to the synthesis problem with limited communication ranges. By utilizing the state saturation, we propose a nonlinear consensus algorithm that solves the consensus problem while preserving the connectedness and the input constraints.