The intercalated disk (ID) is a specialized subcellular region that provides electrical and mechanical connections between myocytes in the heart. The ID has a clearly defined passive role in cardiac tissue, transmitting mechanical forces and electrical currents between cells. Recent studies have shown that Na+ channels, the primary current responsible for cardiac excitation, are preferentially localized at the ID, particularly within nanodomains such as the gap junction–adjacent perinexus and mechanical junction–associated adhesion-excitability nodes, and that perturbations of ID structure alter cardiac conduction. This suggests that the ID may play an important, active role in regulating conduction. However, the structures of the ID and intercellular cleft are not well characterized and, to date, no models have incorporated the influence of ID structure on conduction in cardiac tissue. In this study, we developed an approach to generate realistic finite element model (FEM) meshes replicating nanoscale of the ID structure, based on experimental measurements from transmission electron microscopy images. We then integrated measurements of the intercellular cleft electrical conductivity, derived from the FEM meshes, into a novel cardiac tissue model formulation. FEM-based calculations predict that the distribution of cleft conductances is sensitive to regional changes in ID structure, specifically the intermembrane separation and gap junction distribution. Tissue-scale simulations predict that ID structural heterogeneity leads to significant spatial variation in electrical polarization within the intercellular cleft. Importantly, we found that this heterogeneous cleft polarization regulates conduction by desynchronizing the activation of postjunctional Na+ currents. Additionally, these heterogeneities lead to a weaker dependence of conduction velocity on gap junctional coupling, compared with prior modeling formulations that neglect or simplify ID structure. Further, we found that disruption of local ID nanodomains can either slow or enhance conduction, depending on gap junctional coupling strength. Our study therefore suggests that ID nanoscale structure can play a significant role in regulating cardiac conduction.