As the conversation around using geoengineering to combat climate change intensifies, it is imperative to engage the public and deeply understand their perspectives on geoengineering research, development, and potential deployment. Through a comprehensive data-driven investigation, this paper explores the types of news that captivate public interest in geoengineering. We delved into 30,773 English-language news articles from the BBC and the New York Times, combined with Google Trends data spanning 2018 to 2022, to explore how public interest in geoengineering fluctuates in response to news coverage of broader climate issues. Using BERT-based topic modeling, sentiment analysis, and time-series regression models, we found that positive sentiment in energy-related news serves as a good predictor of heightened public interest in geoengineering, a trend that persists over time. Our findings suggest that public engagement with geoengineering and climate action is not uniform, with some topics being more potent in shaping interest over time, such as climate news related to energy, disasters, and politics. Understanding these patterns is crucial for scientists, policymakers, and educators aiming to craft effective strategies for engaging with the public and fostering dialogue around emerging climate technologies.