The concept of compositeness is used to quantitatively discuss the hadronic molecular nature in exotic hadrons. In this work, we develop a formulation to explicitly introduce the compact quark state contribution in the hadronic potentials together with the direct four-point interaction. We derive analytic expressions of the compositeness from the effective potential between hadrons when the system has a bound state. Applying this formulation to the $X(3872)$, we examine the variation of the compositeness with respect to the binding energy, energy of $\chi_{c1}(2P)$, cutoff momentum, and strength of the direct $D^0\bar{D}^{*0}$ interaction.