The research examines the elements and factors affecting the sustainability conditions of tenders implemented by universities. The actuality of the topic is justified by the strengthening of the universities’ three missions and the emergence of the sustainability aspect, notably its prominent appearance among strategic goals. This study analyzes Hungarian calls for European Union structural funds over two programming periods, focusing on how the conceptual and conditional system of sustainability is integrated to support the Higher Education Institutions (HEIs)’ evolving innovative and third mission role. Regarding the method, the analysis focused on tender documentation from the Széchenyi 2020 (HRDOP) and the strategic guidelines for Széchenyi Plan Plus (EDIOP Plus) (2014–2027). A qualitative content analysis was performed on seven HRDOP calls. Agusdinata’s Human-Centered Design (HCD) and Shared-Action Learning (SAL) framework were utilized to assess the alignment of the calls’ objectives with sustainable development goals (SDGs), particularly concerning the cooperation of the quadruple/quintuple helix actors. The findings indicate that the HRDOP calls’ target system encompasses the elements of the sustainability framework. While sustainability is often indirectly supported, project conditions serve as guidelines for HEIs to undertake SDG-aligned activities. The subsequent EDIOP Plus priorities further emphasize collaboration and cover all framework elements, though relationships with civil society and the educational mission require increased institutional self-contribution. It can be concluded that defining and highlighting these aspects provides guidance during project planning, enhancing the long-term sustainability of university tenders and regional impact.