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Unconventional hydrocarbon reservoirs in layered formations, such as tight sandstones and shales, are continually being developed. Hydraulic fracturing is a critical technology for the high-efficiency development of hydrocarbon reservoirs. Understanding the stress field and stability of the formation interface is vital to understanding stress propagation, preferably before the growing hydraulic fracture contacts the formation interface. In this study, models are developed for computing the stress field of hydraulic fracture propagation near the formation interface, and the stress fields within and at the two sides of the formation interface are analyzed. Four failure modes of the interface under the impact of hydraulic fracture propagation in its vicinity are identified, and the corresponding failure criteria are proposed. By simulating the magnitude and direction of peak stress at different parameters, the failure mode and stability of the formation interface are analyzed. Results reveal that when the interface strength is weak, the formation interface fails before the growing hydraulic fracture contacts it, and its stability is significantly related to a variety of factors, including the type of formation interface, rock mechanical properties, far-field stress, structural parameters, distance between the hydraulic fracture and formation interface, and fracturing execution parameters.