For the first time, this position paper introduces a fundamental link between tensor networks (TNs) and Green AI, highlighting the synergistic potential they hold to enhance both the inclusivity and sustainability of AI research. We argue that TNs are a valuable asset for Green AI, due to their strong mathematical backbone and inherent logarithmic compression potential. In order to demonstrate the significance of establishing the link between Green AI and TNs, we undertake a comprehensive review of the ongoing discussions on Green AI, emphasizing the importance of sustainability and inclusivity in AI research.To support our position, we first provide a comprehensive overview of efficiency metrics proposed in Green AI literature and then evaluate examples of TNs in the fields of kernel machines and deep learning using the proposed efficiency metrics. This position paper aims to incentivize meaningful, constructive discussions by bridging the fundamental principles of Green AI and TNs. We advocate for researchers to seriously evaluate the integration of TNs into their research projects and, in alignment with the link established in this paper, we support prior calls encouraging researchers to treat Green AI principles as a research priority.