The interval function observer designed to reconstructs only a linear function of the state variable. This specialized structure significantly reduces the order and complexity of the traditional state observer, making it particularly suitable for interval estimation of states—an increasingly prominent topic in control theory. First, unlike the interval state observer, the interval function observer is formally defined and constructed for nonlinear interconnected systems that exhibit discrete or continuous state changes over time. Second, sufficient conditions for the existence of such an observer are derived. Using the solution method for the generalized Sylvester equation, the gain matrix for nonlinear interconnected systems is calculated. The design method is further extended from simple systems to more complex interconnected systems. Finally, two models of interconnected systems are introduced, and the proposed design method is validated through two numerical examples, confirming its effectiveness and feasibility.