Software and computational tools are instrumental for scientific investigation in today's digitized research environment. Despite this crucial role, the path towards implementing best practices to achieve reproducibility and sustainability of research software is challenging. Delft University of Technology has begun recently a novel initiative of data stewardship - disciplinary support for research data management, one of the main aims of which is achieving reproducibility of scientific results in general. In this paper, we aim to explore the potential of data stewardship for supporting software reproducibility and sustainability as well. Recently, we gathered the key stakeholders of the topic (i.e. researchers, research software engineers, and data stewards) in a workshop setting to understand the challenges and barriers, the support required to achieve software sustainability and reproducibility, and how all the three parties can efficiently work together. Based on the insights from the workshop, as well as our professional experience as data stewards, we draw conclusions on possible ways forward to achieve the important goal of software reproducibility and sustainability through coordinated efforts of the key stakeholders.