Lithium-ion (Li-ion) battery faults are potentially hazardous to battery health, safety and performance. Thermal fault mechanisms represent a critical subset of such failures. To ensure safety and reliability, battery management systems must have the capability of diagnosing these thermal failures. We present a Partial Differential Equation (PDE) model-based scheme for diagnosing thermal faults in Li-ion batteries. For this study, we adopt a distributed parameter one-dimensional thermal model for cylindrical battery cells. The diagnostic scheme objective is to detect and estimate the size of the thermal fault. The scheme consists of two PDE observers arranged in cascade. The first observer, denoted as Robust Observer, estimates the distributed temperature inside the cell under nominal (healthy) and faulty conditions. The second observer, denoted as Diagnostic Observer, receives this estimated temperature distribution, and in turn outputs a residual signal that provides the fault information. Lyapunov stability theory has been utilized to verify the analytical convergence of the observers under heathy and faulty conditions. Simulation studies are presented to illustrate the effectiveness of the scheme.