The outbreak of the COVID-19 pandemic has been accompanied by a bulk of scientific research and related Twitter discussions. To unravel the public concerns about the COVID-19 crisis reflected in the science-based Twitter conversations, this study tracked the Twitter attention around the COVID-19 research efforts during the first three months of 2020. On the basis of nearly 1.4 million Twitter mentions of 6,162 COVID-19-related scientific publications, we investigated the temporal tweeting dynamic and the Twitter users involved in the online discussions around COVID-19-related research. The results show that the quantity of Twitter mentions of COVID-19-related publications was on rising. Scholarly-oriented Twitter users played an influential role in disseminating research outputs on COVID-19, with their tweets being frequently retweeted. Over time, a change in the focus of the Twitter discussions can be observed, from the initial attention to virological and clinical research to more practical topics, such as the potential treatments, the countermeasures by the governments, the healthcare measures, and the influences on the economy and society, in more recent times.