This work develops a framework for studying the behavior of a passive scalar field in incompressible wall-bounded turbulence using the resolvent operator. This approach expresses the state of the system as the result of applying a linear (resolvent) operator to the nonlinear terms in the governing Navier-Stokes equations. By augmenting the system with a passive scalar equation, this formulation is used to study the relationship between velocity and scalar fluctuations. Additional insight into the mechanisms responsible for driving scalar fluctuations is attained by considering the resolvent form of the passive scalar equation in isolation from the momentum equations. We demonstrate that the passive scalar resolvent operator admits rescaling properties that relates the behavior of scalar fields with different diffusivities, and investigate the ability of this modeling framework to predict statistical properties of the fluctuating scalar field.