The neural substrates of children’s reading and arithmetic skills have long been of great interest to cognitive neuroscientists. However, most previous studies have focused on the contrast between these skills as specific domains. Here, we investigate the potentially shared processes across these domains by focusing on how the neural circuits associated with cognitive control influence reading and arithmetic proficiency in 8-to-10-year-old children. Using a task-free resting state approach, we correlated the intrinsic functional connectivity of the right anterior insula (rAI) network with performance on assessments of Chinese character recognition, reading comprehension, subtraction, and multiplication performance. A common rAI network emerged for reading and arithmetic skill, including bilateral middle temporal gyri (MTG) and superior temporal gyri (STG) in the lateral temporal cortex, as well as inferior frontal gyri (IFG). In addition to these common substrates, performance measures evidenced rAI network specializations. Tasks requiring higher level of cognitive control, such as subtraction and reading comprehension, were associated with hyper connectivity between rAI and dorsoanterior cingulate cortex (dACC) and local insula connectivity. Reading comprehension only, rather than character recognition, was associated with connectivity to IFG, MTG and angular gyrus. Furthermore, subtraction was associated with connectivity to premotor cortex whereas multiplication was associated with the supramarginal gyrus. These results indicate that during a critical period for children’s acquisition of reading and arithmetic, these skills are supported by both intra-network synchronization and inter-network connectivity of rAI circuits. Domain-general intrinsic insular connectivity at rest contained also, functional components that segregated into different sets of skill-related networks. The embedded components of cognitive control may be essential to understanding the dynamic interplay of multiple functional circuits necessary to more fully characterize cognitive skill acquisition.