Superconducting magnets are widely used in accelerator science applications. Muon production solenoids are applications that have recently attracted considerable public attention, after the approval of muon-related physics projects such as coherent muon to electron transition or muon-to-electron-conversion experiments. Based on its characteristics, muon production solenoids tend to be subjected to high radiation exposure, which results in a high heat load being applied to the solenoid magnet, thus limiting the superconducting magnet operation, especially for low-temperature superconductors such as niobium titanium alloy. However, the use of high-temperature superconductors may extend the operation capabilities owing to their functionality at higher temperatures. This study reviews the characteristics of high temperature superconductor magnets in high-radiation environments and their potential for application to muon production solenoids.