Multilayer diffusion-reaction problems appear commonly in heat and mass transfer analysis, including in thermal management of Li-ion batteries, drug delivery and reacting systems. Thermal stability of diffusion-reaction problems is of much interest due to the potential for thermal runaway in such systems. While there is some past work on investigating thermal instability in a one-dimensional multilayer body, practical systems such as Li-ion cells are often two- or even three-dimensional. Therefore, it is important to develop a theoretical understanding of when a two-dimensional multilayer diffusion-reaction system may be thermally unstable. This work addresses this important question by deriving a solution for the transient temperature field in this problem, followed by pole analysis in the Laplace domain. An explicit threshold for thermal stability of the system is determined on the basis of the determinants associated with a set of algebraic equations. The impact of key non-dimensional numbers on thermal stability is determined, including those associated with heat generation and boundary cooling, as well as ratios of thermal properties of the layers. Good agreement with past papers for special cases of the general problem considered here is shown. A stability curve that separates stable and unstable regions in the thermal design space of a two-dimensional two-layered body, such as a stack of Li-ion cells is presented. This work extends the state-of-the-art in the theoretical understanding of thermal stability, and may also contribute towards design for safety of practical engineering systems.