The circadian timing system and integrated stress response (ISR) systems are fundamental regulatory mechanisms that maintain body homeostasis. The central circadian pacemaker in the suprachiasmatic nucleus (SCN) governs daily rhythms through interactions with peripheral oscillators via the hypothalamus-pituitary-adrenal (HPA) axis. On the other hand, ISR signaling is pivotal for preserving cellular homeostasis in response to physiological changes. Notably, disrupted circadian rhythms are observed in cases of impaired ISR signaling. In this work, we examine the potential interplay between the central circadian system and the ISR, mainly through the SCN and HPA axis. We introduce a semimechanistic mathematical model to delineate SCN's capacity for indirectly perceiving physiological stress through glucocorticoid-mediated feedback from the HPA axis and orchestrating a cellular response via the ISR mechanism. Key components of our investigation include evaluating general control nonderepressible 2 (GCN2) expression in the SCN, the effect of physiological stress stimuli on the HPA axis, and the interconnected feedback between the HPA and SCN. Simulation revealed a critical role for GCN2 in linking ISR with circadian rhythms. Experimental findings have demonstrated that a deletion in mice leads to rapid re-entrainment of the circadian clock following jetlag as well as to an elongation of the circadian period. These phenomena are well replicated by our model, which suggests that both the swift re-entrainment and prolonged period can be ascribed to a reduced robustness in neuronal oscillators. Our model also offers insights into phase shifts induced by acute physiological stress and the alignment/misalignment of physiological stress with external light-dark cues. Such understanding aids in strategizing responses to stressful events, such as nutritional status changes and jetlag. This study is the first theoretical work to investigate the complex interaction between integrated stress response (ISR) sensing and central circadian rhythm regulation, encompassing the suprachiasmatic nucleus (SCN) and hypothalamus-pituitary-adrenal (HPA) axis. The findings carry implications for the development of dietary or pharmacological interventions aimed at facilitating recovery from stressful events, such as jetlag. Moreover, they provide promising prospects for potential therapeutic interventions that target circadian rhythm disruption and various stress-related disorders.