After the first weeks of vaccination against SARS-CoV-2, several cases of acute thrombosis were reported. These news reports began to be shared frequently across social media platforms. The aim of this study was to conduct an analysis of Twitter data related to the overall discussion. The data were retrieved from 14 March to 14 April 2021 using the keyword ‘blood clots’. A dataset with n = 266,677 tweets was retrieved, and a systematic random sample of 5% of tweets (n = 13,334) was entered into NodeXL for further analysis. Social network analysis was used to analyse the data by drawing upon the Clauset–Newman–Moore algorithm. Influential users were identified by drawing upon the betweenness centrality measure. Text analysis was applied to identify the key hashtags and websites used at this time. More than half of the network comprised retweets, and the largest groups within the network were broadcast clusters in which a number of key users were retweeted. The most popular narratives involved highlighting the low risk of obtaining a blood clot from a vaccine and highlighting that a number of commonly consumed medicine have higher blood clot risks. A wide variety of users drove the discussion on Twitter, including writers, physicians, the general public, academics, celebrities, and journalists. Twitter was used to highlight the low potential of developing a blood clot from vaccines, and users on Twitter encouraged vaccinations among the public